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Theorem funcsetcestrclem4 18156

Description: Lemma 4 for funcsetcestrc 18162. (Contributed by AV, 27-Mar-2020.)

**Hypotheses**
Ref	Expression
funcsetcestrc.s	⊢ 𝑆 = (SetCat‘𝑈)
funcsetcestrc.c	⊢ 𝐶 = (Base‘𝑆)
funcsetcestrc.f	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))
funcsetcestrc.u	⊢ (𝜑 → 𝑈 ∈ WUni)
funcsetcestrc.o	⊢ (𝜑 → ω ∈ 𝑈)
funcsetcestrc.g	⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑_m 𝑥))))

**Assertion**
Ref	Expression
funcsetcestrclem4	⊢ (𝜑 → 𝐺 Fn (𝐶 × 𝐶))

Distinct variable groups: 𝑥,𝐶 𝜑,𝑥 𝑦,𝐶,𝑥 Allowed substitution hints: 𝜑(𝑦) 𝑆(𝑥,𝑦) 𝑈(𝑥,𝑦) 𝐹(𝑥,𝑦) 𝐺(𝑥,𝑦)

**Proof of Theorem funcsetcestrclem4**
Step	Hyp	Ref	Expression
1		eqid 2728	. . 3 ⊢ (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑_m 𝑥))) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑_m 𝑥)))
2		ovex 7459	. . . 4 ⊢ (𝑦 ↑_m 𝑥) ∈ V
3		resiexg 7926	. . . 4 ⊢ ((𝑦 ↑_m 𝑥) ∈ V → ( I ↾ (𝑦 ↑_m 𝑥)) ∈ V)
4	2, 3	ax-mp 5	. . 3 ⊢ ( I ↾ (𝑦 ↑_m 𝑥)) ∈ V
5	1, 4	fnmpoi 8080	. 2 ⊢ (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑_m 𝑥))) Fn (𝐶 × 𝐶)
6		funcsetcestrc.g	. . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑_m 𝑥))))
7	6	fneq1d 6652	. 2 ⊢ (𝜑 → (𝐺 Fn (𝐶 × 𝐶) ↔ (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑_m 𝑥))) Fn (𝐶 × 𝐶)))
8	5, 7	mpbiri 257	1 ⊢ (𝜑 → 𝐺 Fn (𝐶 × 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3473 {csn 4632 ⟨cop 4638 ↦ cmpt 5235 I cid 5579 × cxp 5680 ↾ cres 5684 Fn wfn 6548 ‘cfv 6553 (class class class)co 7426 ∈ cmpo 7428 ωcom 7876 ↑_m cmap 8851 WUnicwun 10731 ndxcnx 17169 Basecbs 17187 SetCatcsetc 18071

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000

This theorem is referenced by: funcsetcestrc 18162

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