relexpfld - Metamath Proof Explorer

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Theorem relexpfld 15054

Description: The field of an exponentiation of a relation a subset of the relation's field. (Contributed by RP, 23-May-2020.)

**Assertion**
Ref	Expression
relexpfld	⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉) → ∪ ∪ (𝑅↑_𝑟𝑁) ⊆ ∪ ∪ 𝑅)

**Proof of Theorem relexpfld**
Step	Hyp	Ref	Expression
1		simpl 481	. . . . . . . 8 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → 𝑁 = 1)
2	1	oveq2d 7440	. . . . . . 7 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → (𝑅↑_𝑟𝑁) = (𝑅↑_𝑟1))
3		relexp1g 15031	. . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑_𝑟1) = 𝑅)
4	3	ad2antll 727	. . . . . . 7 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → (𝑅↑_𝑟1) = 𝑅)
5	2, 4	eqtrd 2766	. . . . . 6 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → (𝑅↑_𝑟𝑁) = 𝑅)
6	5	unieqd 4926	. . . . 5 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ (𝑅↑_𝑟𝑁) = ∪ 𝑅)
7	6	unieqd 4926	. . . 4 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ ∪ (𝑅↑_𝑟𝑁) = ∪ ∪ 𝑅)
8		eqimss 4038	. . . 4 ⊢ (∪ ∪ (𝑅↑_𝑟𝑁) = ∪ ∪ 𝑅 → ∪ ∪ (𝑅↑_𝑟𝑁) ⊆ ∪ ∪ 𝑅)
9	7, 8	syl 17	. . 3 ⊢ ((𝑁 = 1 ∧ (𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉)) → ∪ ∪ (𝑅↑_𝑟𝑁) ⊆ ∪ ∪ 𝑅)
10	9	ex 411	. 2 ⊢ (𝑁 = 1 → ((𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉) → ∪ ∪ (𝑅↑_𝑟𝑁) ⊆ ∪ ∪ 𝑅))
11		simp2 1134	. . . . . . 7 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ ℕ₀)
12		simp3 1135	. . . . . . 7 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ 𝑉)
13		simp1 1133	. . . . . . . 8 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉) → ¬ 𝑁 = 1)
14	13	pm2.21d 121	. . . . . . 7 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉) → (𝑁 = 1 → Rel 𝑅))
15	11, 12, 14	3jca 1125	. . . . . 6 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉) → (𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)))
16		relexprelg 15043	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉 ∧ (𝑁 = 1 → Rel 𝑅)) → Rel (𝑅↑_𝑟𝑁))
17		relfld 6286	. . . . . 6 ⊢ (Rel (𝑅↑_𝑟𝑁) → ∪ ∪ (𝑅↑_𝑟𝑁) = (dom (𝑅↑_𝑟𝑁) ∪ ran (𝑅↑_𝑟𝑁)))
18	15, 16, 17	3syl 18	. . . . 5 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉) → ∪ ∪ (𝑅↑_𝑟𝑁) = (dom (𝑅↑_𝑟𝑁) ∪ ran (𝑅↑_𝑟𝑁)))
19		elnn0 12526	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0))
20		relexpnndm 15046	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑_𝑟𝑁) ⊆ dom 𝑅)
21		relexpnnrn 15050	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑_𝑟𝑁) ⊆ ran 𝑅)
22		unss12 4183	. . . . . . . . . 10 ⊢ ((dom (𝑅↑_𝑟𝑁) ⊆ dom 𝑅 ∧ ran (𝑅↑_𝑟𝑁) ⊆ ran 𝑅) → (dom (𝑅↑_𝑟𝑁) ∪ ran (𝑅↑_𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅))
23	20, 21, 22	syl2anc 582	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑅 ∈ 𝑉) → (dom (𝑅↑_𝑟𝑁) ∪ ran (𝑅↑_𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅))
24	23	ex 411	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → (𝑅 ∈ 𝑉 → (dom (𝑅↑_𝑟𝑁) ∪ ran (𝑅↑_𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅)))
25		simpl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → 𝑁 = 0)
26	25	oveq2d 7440	. . . . . . . . . . . . . 14 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑_𝑟𝑁) = (𝑅↑_𝑟0))
27		relexp0g 15027	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑_𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
28	27	adantl 480	. . . . . . . . . . . . . 14 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑_𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
29	26, 28	eqtrd 2766	. . . . . . . . . . . . 13 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (𝑅↑_𝑟𝑁) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
30	29	dmeqd 5912	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑_𝑟𝑁) = dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
31		dmresi 6061	. . . . . . . . . . . 12 ⊢ dom ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
32	30, 31	eqtrdi 2782	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑_𝑟𝑁) = (dom 𝑅 ∪ ran 𝑅))
33		eqimss 4038	. . . . . . . . . . 11 ⊢ (dom (𝑅↑_𝑟𝑁) = (dom 𝑅 ∪ ran 𝑅) → dom (𝑅↑_𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
34	32, 33	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → dom (𝑅↑_𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
35	29	rneqd 5944	. . . . . . . . . . . 12 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑_𝑟𝑁) = ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
36		rnresi 6084	. . . . . . . . . . . 12 ⊢ ran ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (dom 𝑅 ∪ ran 𝑅)
37	35, 36	eqtrdi 2782	. . . . . . . . . . 11 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑_𝑟𝑁) = (dom 𝑅 ∪ ran 𝑅))
38		eqimss 4038	. . . . . . . . . . 11 ⊢ (ran (𝑅↑_𝑟𝑁) = (dom 𝑅 ∪ ran 𝑅) → ran (𝑅↑_𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
39	37, 38	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → ran (𝑅↑_𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
40	34, 39	unssd 4187	. . . . . . . . 9 ⊢ ((𝑁 = 0 ∧ 𝑅 ∈ 𝑉) → (dom (𝑅↑_𝑟𝑁) ∪ ran (𝑅↑_𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅))
41	40	ex 411	. . . . . . . 8 ⊢ (𝑁 = 0 → (𝑅 ∈ 𝑉 → (dom (𝑅↑_𝑟𝑁) ∪ ran (𝑅↑_𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅)))
42	24, 41	jaoi 855	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑅 ∈ 𝑉 → (dom (𝑅↑_𝑟𝑁) ∪ ran (𝑅↑_𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅)))
43	19, 42	sylbi 216	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → (𝑅 ∈ 𝑉 → (dom (𝑅↑_𝑟𝑁) ∪ ran (𝑅↑_𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅)))
44	11, 12, 43	sylc 65	. . . . 5 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉) → (dom (𝑅↑_𝑟𝑁) ∪ ran (𝑅↑_𝑟𝑁)) ⊆ (dom 𝑅 ∪ ran 𝑅))
45	18, 44	eqsstrd 4018	. . . 4 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉) → ∪ ∪ (𝑅↑_𝑟𝑁) ⊆ (dom 𝑅 ∪ ran 𝑅))
46		dmrnssfld 5977	. . . 4 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅
47	45, 46	sstrdi 3992	. . 3 ⊢ ((¬ 𝑁 = 1 ∧ 𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉) → ∪ ∪ (𝑅↑_𝑟𝑁) ⊆ ∪ ∪ 𝑅)
48	47	3expib 1119	. 2 ⊢ (¬ 𝑁 = 1 → ((𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉) → ∪ ∪ (𝑅↑_𝑟𝑁) ⊆ ∪ ∪ 𝑅))
49	10, 48	pm2.61i 182	1 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ 𝑉) → ∪ ∪ (𝑅↑_𝑟𝑁) ⊆ ∪ ∪ 𝑅)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∨ wo 845 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ∪ cun 3945 ⊆ wss 3947 ∪ cuni 4913 I cid 5579 dom cdm 5682 ran crn 5683 ↾ cres 5684 Rel wrel 5687 (class class class)co 7424 0cc0 11158 1c1 11159 ℕcn 12264 ℕ₀cn0 12524 ↑_𝑟crelexp 15024

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 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-n0 12525 df-z 12611 df-uz 12875 df-seq 14022 df-relexp 15025

This theorem is referenced by: relexpfldd 15055

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