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Theorem funcestrcsetclem5 17510

Description: Lemma 5 for funcestrcsetc 17515. (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 484	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑_m (Base‘𝑥)))))
3		fveq2 6674	. . . . . 6 ⊢ (𝑦 = 𝑌 → (Base‘𝑦) = (Base‘𝑌))
4		fveq2 6674	. . . . . 6 ⊢ (𝑥 = 𝑋 → (Base‘𝑥) = (Base‘𝑋))
5	3, 4	oveqan12rd 7190	. . . . 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 7182	. . . . 5 ⊢ (𝑁 ↑_m 𝑀) = ((Base‘𝑌) ↑_m (Base‘𝑋))
9	5, 8	eqtr4di 2791	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((Base‘𝑦) ↑_m (Base‘𝑥)) = (𝑁 ↑_m 𝑀))
10	9	reseq2d 5825	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ( I ↾ ((Base‘𝑦) ↑_m (Base‘𝑥))) = ( I ↾ (𝑁 ↑_m 𝑀)))
11	10	adantl 485	. 2 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ( I ↾ ((Base‘𝑦) ↑_m (Base‘𝑥))) = ( I ↾ (𝑁 ↑_m 𝑀)))
12		simprl 771	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
13		simprr 773	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ 𝐵)
14		ovexd 7205	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑁 ↑_m 𝑀) ∈ V)
15	14	resiexd 6989	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ( I ↾ (𝑁 ↑_m 𝑀)) ∈ V)
16	2, 11, 12, 13, 15	ovmpod 7317	1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑁 ↑_m 𝑀)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 Vcvv 3398 ↦ cmpt 5110 I cid 5428 ↾ cres 5527 ‘cfv 6339 (class class class)co 7170 ∈ cmpo 7172 ↑_m cmap 8437 WUnicwun 10200 Basecbs 16586 SetCatcsetc 17447 ExtStrCatcestrc 17488

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5296

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-ov 7173 df-oprab 7174 df-mpo 7175

This theorem is referenced by: funcestrcsetclem6 17511 funcestrcsetclem7 17512 funcestrcsetclem8 17513 funcestrcsetclem9 17514

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