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Theorem funcestrcsetclem6 18157

Description: Lemma 6 for funcestrcsetc 18161. (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
funcestrcsetclem6	⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐻 ∈ (𝑁 ↑_m 𝑀)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)

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) 𝐻(𝑥,𝑦)

**Proof of Theorem funcestrcsetclem6**
Step	Hyp	Ref	Expression
1		funcestrcsetc.e	. . . . 5 ⊢ 𝐸 = (ExtStrCat‘𝑈)
2		funcestrcsetc.s	. . . . 5 ⊢ 𝑆 = (SetCat‘𝑈)
3		funcestrcsetc.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐸)
4		funcestrcsetc.c	. . . . 5 ⊢ 𝐶 = (Base‘𝑆)
5		funcestrcsetc.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ WUni)
6		funcestrcsetc.f	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))
7		funcestrcsetc.g	. . . . 5 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑_m (Base‘𝑥)))))
8		funcestrcsetc.m	. . . . 5 ⊢ 𝑀 = (Base‘𝑋)
9		funcestrcsetc.n	. . . . 5 ⊢ 𝑁 = (Base‘𝑌)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	funcestrcsetclem5 18156	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑁 ↑_m 𝑀)))
11	10	3adant3 1132	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐻 ∈ (𝑁 ↑_m 𝑀)) → (𝑋𝐺𝑌) = ( I ↾ (𝑁 ↑_m 𝑀)))
12	11	fveq1d 6878	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐻 ∈ (𝑁 ↑_m 𝑀)) → ((𝑋𝐺𝑌)‘𝐻) = (( I ↾ (𝑁 ↑_m 𝑀))‘𝐻))
13		fvresi 7165	. . 3 ⊢ (𝐻 ∈ (𝑁 ↑_m 𝑀) → (( I ↾ (𝑁 ↑_m 𝑀))‘𝐻) = 𝐻)
14	13	3ad2ant3 1135	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐻 ∈ (𝑁 ↑_m 𝑀)) → (( I ↾ (𝑁 ↑_m 𝑀))‘𝐻) = 𝐻)
15	12, 14	eqtrd 2770	1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐻 ∈ (𝑁 ↑_m 𝑀)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ↦ cmpt 5201 I cid 5547 ↾ cres 5656 ‘cfv 6531 (class class class)co 7405 ∈ cmpo 7407 ↑_m cmap 8840 WUnicwun 10714 Basecbs 17228 SetCatcsetc 18088 ExtStrCatcestrc 18134

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pr 5402

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-ov 7408 df-oprab 7409 df-mpo 7410

This theorem is referenced by: funcestrcsetclem9 18160 fthestrcsetc 18162 fullestrcsetc 18163

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