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Theorem funcestrcsetclem5 18068

Description: Lemma 5 for funcestrcsetc 18073. (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 480	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑_m (Base‘𝑥)))))
3		fveq2 6826	. . . . . 6 ⊢ (𝑦 = 𝑌 → (Base‘𝑦) = (Base‘𝑌))
4		fveq2 6826	. . . . . 6 ⊢ (𝑥 = 𝑋 → (Base‘𝑥) = (Base‘𝑋))
5	3, 4	oveqan12rd 7373	. . . . 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 7365	. . . . 5 ⊢ (𝑁 ↑_m 𝑀) = ((Base‘𝑌) ↑_m (Base‘𝑋))
9	5, 8	eqtr4di 2782	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((Base‘𝑦) ↑_m (Base‘𝑥)) = (𝑁 ↑_m 𝑀))
10	9	reseq2d 5934	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ( I ↾ ((Base‘𝑦) ↑_m (Base‘𝑥))) = ( I ↾ (𝑁 ↑_m 𝑀)))
11	10	adantl 481	. 2 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ( I ↾ ((Base‘𝑦) ↑_m (Base‘𝑥))) = ( I ↾ (𝑁 ↑_m 𝑀)))
12		simprl 770	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
13		simprr 772	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ 𝐵)
14		ovexd 7388	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑁 ↑_m 𝑀) ∈ V)
15	14	resiexd 7156	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ( I ↾ (𝑁 ↑_m 𝑀)) ∈ V)
16	2, 11, 12, 13, 15	ovmpod 7505	1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑁 ↑_m 𝑀)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 Vcvv 3438 ↦ cmpt 5176 I cid 5517 ↾ cres 5625 ‘cfv 6486 (class class class)co 7353 ∈ cmpo 7355 ↑_m cmap 8760 WUnicwun 10613 Basecbs 17138 SetCatcsetc 18000 ExtStrCatcestrc 18046

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pr 5374

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 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4479 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5518 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-ov 7356 df-oprab 7357 df-mpo 7358

This theorem is referenced by: funcestrcsetclem6 18069 funcestrcsetclem7 18070 funcestrcsetclem8 18071 funcestrcsetclem9 18072

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