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Theorem funcestrcsetclem5 18233

Description: Lemma 5 for funcestrcsetc 18238. (Contributed by AV, 23-Mar-2020.)

**Hypotheses**
Ref	Expression
funcestrcsetc.e	⊢ 𝐸 = (ExtStrCat‘𝑈)
funcestrcsetc.s	⊢ 𝑆 = (SetCat‘𝑈)
funcestrcsetc.b	⊢ 𝐵 = (Base‘𝐸)
funcestrcsetc.c	⊢ 𝐶 = (Base‘𝑆)
funcestrcsetc.u	⊢ (𝜑 → 𝑈 ∈ WUni)
funcestrcsetc.f	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))
funcestrcsetc.g	⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑_m (Base‘𝑥)))))
funcestrcsetc.m	⊢ 𝑀 = (Base‘𝑋)
funcestrcsetc.n	⊢ 𝑁 = (Base‘𝑌)

**Assertion**
Ref	Expression
funcestrcsetclem5	⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑁 ↑_m 𝑀)))

Distinct variable groups: 𝑥,𝐵 𝑥,𝑋 𝜑,𝑥 𝑥,𝐶 𝑦,𝐵,𝑥 𝑦,𝑋 𝜑,𝑦 𝑥,𝑀,𝑦 𝑥,𝑁,𝑦 𝑥,𝑌,𝑦 Allowed substitution hints: 𝐶(𝑦) 𝑆(𝑥, 𝑦) 𝑈(𝑥, 𝑦) 𝐸(𝑥, 𝑦) 𝐹(𝑥, 𝑦) 𝐺(𝑥, 𝑦)

**Proof of Theorem funcestrcsetclem5**
Step	Hyp	Ref	Expression
1		funcestrcsetc.g	. . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑_m (Base‘𝑥)))))
2	1	adantr 486	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑_m (Base‘𝑥)))))
3		fveq2 6879	. . . . . 6 ⊢ (𝑦 = 𝑌 → (Base‘𝑦) = (Base‘𝑌))
4		fveq2 6879	. . . . . 6 ⊢ (𝑥 = 𝑋 → (Base‘𝑥) = (Base‘𝑋))
5	3, 4	oveqan12rd 7434	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((Base‘𝑦) ↑_m (Base‘𝑥)) = ((Base‘𝑌) ↑_m (Base‘𝑋)))
6		funcestrcsetc.n	. . . . . 6 ⊢ 𝑁 = (Base‘𝑌)
7		funcestrcsetc.m	. . . . . 6 ⊢ 𝑀 = (Base‘𝑋)
8	6, 7	oveq12i 7426	. . . . 5 ⊢ (𝑁 ↑_m 𝑀) = ((Base‘𝑌) ↑_m (Base‘𝑋))
9	5, 8	eqtr4di 2813	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((Base‘𝑦) ↑_m (Base‘𝑥)) = (𝑁 ↑_m 𝑀))
10	9	reseq2d 5972	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ( I ↾ ((Base‘𝑦) ↑_m (Base‘𝑥))) = ( I ↾ (𝑁 ↑_m 𝑀)))
11	10	adantl 487	. 2 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ( I ↾ ((Base‘𝑦) ↑_m (Base‘𝑥))) = ( I ↾ (𝑁 ↑_m 𝑀)))
12		simprl 783	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
13		simprr 785	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ 𝐵)
14		ovexd 7449	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑁 ↑_m 𝑀) ∈ V)
15	14	resiexd 7216	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ( I ↾ (𝑁 ↑_m 𝑀)) ∈ V)
16	2, 11, 12, 13, 15	ovmpod 7566	1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑁 ↑_m 𝑀)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 Vcvv 3450 ↦ cmpt 5186 I cid 5549 ↾ cres 5657 ‘cfv 6533 (class class class)co 7414 ∈ cmpo 7416 ↑_m cmap 8827 WUnicwun 10710 Basecbs 17302 SetCatcsetc 18165 ExtStrCatcestrc 18211

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pr 5398

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7417 df-oprab 7418 df-mpo 7419

This theorem is used by: funcestrcsetclem6 18234 funcestrcsetclem7 18235 funcestrcsetclem8 18236 funcestrcsetclem9 18237

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