Theorem ackbij1lem9 10251

Description: Lemma for ackbij1 10261. (Contributed by Stefan O'Rear, 19-Nov-2014.)

Hypothesis

Ref	Expression
ackbij.f	⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))

Assertion

Ref	Expression
ackbij1lem9	⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹‘(𝐴 ∪ 𝐵)) = ((𝐹‘𝐴) +o (𝐹‘𝐵)))

Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Proof of Theorem ackbij1lem9

Step	Hyp	Ref	Expression
1		elinel2 4196	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ Fin)
2	1	3ad2ant1 1131	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ∈ Fin)
3		snfi 9068	. . . . . . . . . 10 ⊢ {𝑦} ∈ Fin
4		elinel1 4195	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ 𝒫 ω)
5	4	elpwid 4612	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ⊆ ω)
6	5	3ad2ant1 1131	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ⊆ ω)
7		onfin2 9255	. . . . . . . . . . . . . 14 ⊢ ω = (On ∩ Fin)
8		inss2 4230	. . . . . . . . . . . . . 14 ⊢ (On ∩ Fin) ⊆ Fin
9	7, 8	eqsstri 4014	. . . . . . . . . . . . 13 ⊢ ω ⊆ Fin
10	6, 9	sstrdi 3992	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ⊆ Fin)
11	10	sselda 3980	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ Fin)
12		pwfi 9202	. . . . . . . . . . 11 ⊢ (𝑦 ∈ Fin ↔ 𝒫 𝑦 ∈ Fin)
13	11, 12	sylib 217	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ 𝐴) → 𝒫 𝑦 ∈ Fin)
14		xpfi 9341	. . . . . . . . . 10 ⊢ (({𝑦} ∈ Fin ∧ 𝒫 𝑦 ∈ Fin) → ({𝑦} × 𝒫 𝑦) ∈ Fin)
15	3, 13, 14	sylancr 586	. . . . . . . . 9 ⊢ (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ 𝐴) → ({𝑦} × 𝒫 𝑦) ∈ Fin)
16	15	ralrimiva 3143	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ∀𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin)
17		iunfi 9364	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin) → ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin)
18	2, 16, 17	syl2anc 583	. . . . . . 7 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin)
19		ficardid 9985	. . . . . . 7 ⊢ (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin → (card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ≈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦))
20	18, 19	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ≈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦))
21		elinel2 4196	. . . . . . . . 9 ⊢ (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ∈ Fin)
22	21	3ad2ant2 1132	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ∈ Fin)
23		elinel1 4195	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ∈ 𝒫 ω)
24	23	elpwid 4612	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ⊆ ω)
25	24	3ad2ant2 1132	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ⊆ ω)
26	25, 9	sstrdi 3992	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ⊆ Fin)
27	26	sselda 3980	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ Fin)
28	27, 12	sylib 217	. . . . . . . . . 10 ⊢ (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ 𝐵) → 𝒫 𝑦 ∈ Fin)
29	3, 28, 14	sylancr 586	. . . . . . . . 9 ⊢ (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ 𝐵) → ({𝑦} × 𝒫 𝑦) ∈ Fin)
30	29	ralrimiva 3143	. . . . . . . 8 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ∀𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin)
31		iunfi 9364	. . . . . . . 8 ⊢ ((𝐵 ∈ Fin ∧ ∀𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin) → ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin)
32	22, 30, 31	syl2anc 583	. . . . . . 7 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin)
33		ficardid 9985	. . . . . . 7 ⊢ (∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin → (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) ≈ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))
34	32, 33	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) ≈ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))
35		djuen 10192	. . . . . 6 ⊢ (((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ≈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∧ (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) ≈ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) → ((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ⊔ (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))) ≈ (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ⊔ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)))
36	20, 34, 35	syl2anc 583	. . . . 5 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ⊔ (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))) ≈ (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ⊔ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)))
37		djudisj 6171	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) = ∅ → (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∩ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) = ∅)
38	37	3ad2ant3 1133	. . . . . . 7 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∩ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) = ∅)
39		endjudisj 10191	. . . . . . 7 ⊢ ((∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin ∧ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin ∧ (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∩ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) = ∅) → (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ⊔ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) ≈ (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∪ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)))
40	18, 32, 38, 39	syl3anc 1369	. . . . . 6 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ⊔ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) ≈ (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∪ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)))
41		iunxun 5097	. . . . . 6 ⊢ ∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦) = (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∪ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))
42	40, 41	breqtrrdi 5190	. . . . 5 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ⊔ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) ≈ ∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦))
43		entr 9026	. . . . 5 ⊢ ((((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ⊔ (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))) ≈ (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ⊔ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) ∧ (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ⊔ ∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) ≈ ∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦)) → ((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ⊔ (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))) ≈ ∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦))
44	36, 42, 43	syl2anc 583	. . . 4 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ⊔ (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))) ≈ ∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦))
45		carden2b 9990	. . . 4 ⊢ (((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ⊔ (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))) ≈ ∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦) → (card‘((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ⊔ (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)))) = (card‘∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦)))
46	44, 45	syl 17	. . 3 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ⊔ (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)))) = (card‘∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦)))
47		ficardom 9984	. . . . 5 ⊢ (∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin → (card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ∈ ω)
48	18, 47	syl 17	. . . 4 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ∈ ω)
49		ficardom 9984	. . . . 5 ⊢ (∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin → (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) ∈ ω)
50	32, 49	syl 17	. . . 4 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) ∈ ω)
51		nnadju 10220	. . . 4 ⊢ (((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ∈ ω ∧ (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)) ∈ ω) → (card‘((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ⊔ (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)))) = ((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) +o (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))))
52	48, 50, 51	syl2anc 583	. . 3 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) ⊔ (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)))) = ((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) +o (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))))
53	46, 52	eqtr3d 2770	. 2 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (card‘∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦)) = ((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) +o (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))))
54		ackbij1lem6 10248	. . . 4 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → (𝐴 ∪ 𝐵) ∈ (𝒫 ω ∩ Fin))
55	54	3adant3 1130	. . 3 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) ∈ (𝒫 ω ∩ Fin))
56		ackbij.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))
57	56	ackbij1lem7 10249	. . 3 ⊢ ((𝐴 ∪ 𝐵) ∈ (𝒫 ω ∩ Fin) → (𝐹‘(𝐴 ∪ 𝐵)) = (card‘∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦)))
58	55, 57	syl 17	. 2 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹‘(𝐴 ∪ 𝐵)) = (card‘∪ 𝑦 ∈ (𝐴 ∪ 𝐵)({𝑦} × 𝒫 𝑦)))
59	56	ackbij1lem7 10249	. . . 4 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹‘𝐴) = (card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)))
60	56	ackbij1lem7 10249	. . . 4 ⊢ (𝐵 ∈ (𝒫 ω ∩ Fin) → (𝐹‘𝐵) = (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦)))
61	59, 60	oveqan12d 7439	. . 3 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → ((𝐹‘𝐴) +o (𝐹‘𝐵)) = ((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) +o (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))))
62	61	3adant3 1130	. 2 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹‘𝐴) +o (𝐹‘𝐵)) = ((card‘∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝒫 𝑦)) +o (card‘∪ 𝑦 ∈ 𝐵 ({𝑦} × 𝒫 𝑦))))
63	53, 58, 62	3eqtr4d 2778	1 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹‘(𝐴 ∪ 𝐵)) = ((𝐹‘𝐴) +o (𝐹‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  ∀wral 3058   ∪ cun 3945   ∩ cin 3946   ⊆ wss 3947  ∅c0 4323  𝒫 cpw 4603  {csn 4629  ∪ ciun 4996   class class class wbr 5148   ↦ cmpt 5231   × cxp 5676  Oncon0 6369  ‘cfv 6548  (class class class)co 7420  ωcom 7870   +_o coa 8483   ≈ cen 8960  Fincfn 8963   ⊔ cdju 9921  cardccrd 9958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-1o 8486  df-oadd 8490  df-er 8724  df-en 8964  df-dom 8965  df-sdom 8966  df-fin 8967  df-dju 9924  df-card 9962

This theorem is referenced by:  ackbij1lem12  10254  ackbij1lem13  10255  ackbij1lem14  10256  ackbij1lem16  10258  ackbij1lem18  10260

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