Theorem funcestrcsetclem5 17393

Description: Lemma 5 for funcestrcsetc 17398. (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 483	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))
3		fveq2 6669	. . . . . 6 ⊢ (𝑦 = 𝑌 → (Base‘𝑦) = (Base‘𝑌))
4		fveq2 6669	. . . . . 6 ⊢ (𝑥 = 𝑋 → (Base‘𝑥) = (Base‘𝑋))
5	3, 4	oveqan12rd 7175	. . . . 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 7167	. . . . 5 ⊢ (𝑁 ↑m 𝑀) = ((Base‘𝑌) ↑m (Base‘𝑋))
9	5, 8	syl6eqr 2874	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((Base‘𝑦) ↑m (Base‘𝑥)) = (𝑁 ↑m 𝑀))
10	9	reseq2d 5852	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) = ( I ↾ (𝑁 ↑m 𝑀)))
11	10	adantl 484	. 2 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) = ( I ↾ (𝑁 ↑m 𝑀)))
12		simprl 769	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
13		simprr 771	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ 𝐵)
14		ovexd 7190	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑁 ↑m 𝑀) ∈ V)
15	14	resiexd 6978	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ( I ↾ (𝑁 ↑m 𝑀)) ∈ V)
16	2, 11, 12, 13, 15	ovmpod 7301	1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑁 ↑m 𝑀)))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  Vcvv 3494   ↦ cmpt 5145   I cid 5458   ↾ cres 5556  ‘cfv 6354  (class class class)co 7155   ∈ cmpo 7157   ↑_m cmap 8405  WUnicwun 10121  Basecbs 16482  SetCatcsetc 17334  ExtStrCatcestrc 17371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7158  df-oprab 7159  df-mpo 7160

This theorem is referenced by:  funcestrcsetclem6  17394  funcestrcsetclem7  17395  funcestrcsetclem8  17396  funcestrcsetclem9  17397