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Theorem funcestrcsetclem6 18102

Description: Lemma 6 for funcestrcsetc 18106. (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 18101	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑁 ↑_m 𝑀)))
11	10	3adant3 1138	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐻 ∈ (𝑁 ↑_m 𝑀)) → (𝑋𝐺𝑌) = ( I ↾ (𝑁 ↑_m 𝑀)))
12	11	fveq1d 6829	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐻 ∈ (𝑁 ↑_m 𝑀)) → ((𝑋𝐺𝑌)‘𝐻) = (( I ↾ (𝑁 ↑_m 𝑀))‘𝐻))
13		fvresi 7117	. . 3 ⊢ (𝐻 ∈ (𝑁 ↑_m 𝑀) → (( I ↾ (𝑁 ↑_m 𝑀))‘𝐻) = 𝐻)
14	13	3ad2ant3 1141	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐻 ∈ (𝑁 ↑_m 𝑀)) → (( I ↾ (𝑁 ↑_m 𝑀))‘𝐻) = 𝐻)
15	12, 14	eqtrd 2774	1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐻 ∈ (𝑁 ↑_m 𝑀)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ↦ cmpt 5153 I cid 5512 ↾ cres 5620 ‘cfv 6485 (class class class)co 7356 ∈ cmpo 7358 ↑_m cmap 8763 WUnicwun 10614 Basecbs 17170 SetCatcsetc 18033 ExtStrCatcestrc 18079

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 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pr 5362

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-ral 3054 df-rex 3064 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4262 df-if 4455 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-id 5513 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-ov 7359 df-oprab 7360 df-mpo 7361

This theorem is referenced by: funcestrcsetclem9 18105 fthestrcsetc 18107 fullestrcsetc 18108

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