Theorem funcestrcsetclem4 18091

Description: Lemma 4 for funcestrcsetc 18097. (Contributed by AV, 22-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)))))

Assertion

Ref	Expression
funcestrcsetclem4	⊢ (𝜑 → 𝐺 Fn (𝐵 × 𝐵))

Distinct variable groups:   𝑥,𝐵   𝜑,𝑥   𝑥,𝐶   𝑦,𝐵,𝑥

Allowed substitution hints:   𝜑(𝑦)   𝐶(𝑦)   𝑆(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝐸(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem funcestrcsetclem4

Step	Hyp	Ref	Expression
1		eqid 2732	. . 3 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))
2		ovex 7438	. . . 4 ⊢ ((Base‘𝑦) ↑m (Base‘𝑥)) ∈ V
3		resiexg 7901	. . . 4 ⊢ (((Base‘𝑦) ↑m (Base‘𝑥)) ∈ V → ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ∈ V)
4	2, 3	ax-mp 5	. . 3 ⊢ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ∈ V
5	1, 4	fnmpoi 8052	. 2 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))) Fn (𝐵 × 𝐵)
6		funcestrcsetc.g	. . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))
7	6	fneq1d 6639	. 2 ⊢ (𝜑 → (𝐺 Fn (𝐵 × 𝐵) ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))) Fn (𝐵 × 𝐵)))
8	5, 7	mpbiri 257	1 ⊢ (𝜑 → 𝐺 Fn (𝐵 × 𝐵))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  Vcvv 3474   ↦ cmpt 5230   I cid 5572   × cxp 5673   ↾ cres 5677   Fn wfn 6535  ‘cfv 6540  (class class class)co 7405   ∈ cmpo 7407   ↑_m cmap 8816  WUnicwun 10691  Basecbs 17140  SetCatcsetc 18021  ExtStrCatcestrc 18069

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972

This theorem is referenced by:  funcestrcsetc  18097