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Theorem funcestrcsetclem6 18080

Description: Lemma 6 for funcestrcsetc 18084. (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 18079	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑁 ↑_m 𝑀)))
11	10	3adant3 1133	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐻 ∈ (𝑁 ↑_m 𝑀)) → (𝑋𝐺𝑌) = ( I ↾ (𝑁 ↑_m 𝑀)))
12	11	fveq1d 6844	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐻 ∈ (𝑁 ↑_m 𝑀)) → ((𝑋𝐺𝑌)‘𝐻) = (( I ↾ (𝑁 ↑_m 𝑀))‘𝐻))
13		fvresi 7129	. . 3 ⊢ (𝐻 ∈ (𝑁 ↑_m 𝑀) → (( I ↾ (𝑁 ↑_m 𝑀))‘𝐻) = 𝐻)
14	13	3ad2ant3 1136	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐻 ∈ (𝑁 ↑_m 𝑀)) → (( I ↾ (𝑁 ↑_m 𝑀))‘𝐻) = 𝐻)
15	12, 14	eqtrd 2772	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 1087 = wceq 1542 ∈ wcel 2114 ↦ cmpt 5181 I cid 5526 ↾ cres 5634 ‘cfv 6500 (class class class)co 7368 ∈ cmpo 7370 ↑_m cmap 8775 WUnicwun 10623 Basecbs 17148 SetCatcsetc 18011 ExtStrCatcestrc 18057

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pr 5379

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7371 df-oprab 7372 df-mpo 7373

This theorem is referenced by: funcestrcsetclem9 18083 fthestrcsetc 18085 fullestrcsetc 18086

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