Theorem dfco2a 6245

Description: Generalization of dfco2 6244, where 𝐶 can have any value between dom 𝐴 ∩ ran 𝐵 and V. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
dfco2a	⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ 𝐶 ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem dfco2a

Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfco2 6244	. 2 ⊢ (𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
2		vex 3474	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
3	2	eliniseg 6093	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ V → (𝑧 ∈ (◡𝐵 “ {𝑥}) ↔ 𝑧𝐵𝑥))
4	3	elv 3476	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (◡𝐵 “ {𝑥}) ↔ 𝑧𝐵𝑥)
5		vex 3474	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
6	2, 5	brelrn 5939	. . . . . . . . . . . . 13 ⊢ (𝑧𝐵𝑥 → 𝑥 ∈ ran 𝐵)
7	4, 6	sylbi 216	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (◡𝐵 “ {𝑥}) → 𝑥 ∈ ran 𝐵)
8		vex 3474	. . . . . . . . . . . . . 14 ⊢ 𝑤 ∈ V
9	5, 8	elimasn 6088	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑤⟩ ∈ 𝐴)
10	5, 8	opeldm 5905	. . . . . . . . . . . . 13 ⊢ (⟨𝑥, 𝑤⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴)
11	9, 10	sylbi 216	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (𝐴 “ {𝑥}) → 𝑥 ∈ dom 𝐴)
12	7, 11	anim12ci 613	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (◡𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥})) → (𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ ran 𝐵))
13	12	adantl 481	. . . . . . . . . 10 ⊢ ((𝑦 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ (◡𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥}))) → (𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ ran 𝐵))
14	13	exlimivv 1928	. . . . . . . . 9 ⊢ (∃𝑧∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ (◡𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥}))) → (𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ ran 𝐵))
15		elxp 5696	. . . . . . . . 9 ⊢ (𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑧∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ (◡𝐵 “ {𝑥}) ∧ 𝑤 ∈ (𝐴 “ {𝑥}))))
16		elin 3961	. . . . . . . . 9 ⊢ (𝑥 ∈ (dom 𝐴 ∩ ran 𝐵) ↔ (𝑥 ∈ dom 𝐴 ∧ 𝑥 ∈ ran 𝐵))
17	14, 15, 16	3imtr4i 292	. . . . . . . 8 ⊢ (𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) → 𝑥 ∈ (dom 𝐴 ∩ ran 𝐵))
18		ssel 3972	. . . . . . . 8 ⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝑥 ∈ (dom 𝐴 ∩ ran 𝐵) → 𝑥 ∈ 𝐶))
19	17, 18	syl5 34	. . . . . . 7 ⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) → 𝑥 ∈ 𝐶))
20	19	pm4.71rd 562	. . . . . 6 ⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))))
21	20	exbidv 1917	. . . . 5 ⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (∃𝑥 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))))
22		rexv 3496	. . . . 5 ⊢ (∃𝑥 ∈ V 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
23		df-rex 3067	. . . . 5 ⊢ (∃𝑥 ∈ 𝐶 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))))
24	21, 22, 23	3bitr4g 314	. . . 4 ⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (∃𝑥 ∈ V 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 ∈ 𝐶 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))))
25		eliun 4996	. . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 ∈ V 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
26		eliun 4996	. . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐶 ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 ∈ 𝐶 𝑦 ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
27	24, 25, 26	3bitr4g 314	. . 3 ⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝑦 ∈ ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐶 ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))))
28	27	eqrdv 2726	. 2 ⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) = ∪ 𝑥 ∈ 𝐶 ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
29	1, 28	eqtrid 2780	1 ⊢ ((dom 𝐴 ∩ ran 𝐵) ⊆ 𝐶 → (𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ 𝐶 ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534  ∃wex 1774   ∈ wcel 2099  ∃wrex 3066  Vcvv 3470   ∩ cin 3944   ⊆ wss 3945  {csn 4625  ⟨cop 4631  ∪ ciun 4992   class class class wbr 5143   × cxp 5671  ^◡ccnv 5672  dom cdm 5673  ran crn 5674   “ cima 5676   ∘ ccom 5677

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 5294  ax-nul 5301  ax-pr 5424

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-iun 4994  df-br 5144  df-opab 5206  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686

This theorem is referenced by:  fparlem3  8114  fparlem4  8115

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