Theorem funcringcsetcALTV2lem4 44317

Description: Lemma 4 for funcringcsetcALTV2 44323. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)

Hypotheses

Ref	Expression
funcringcsetcALTV2.r	⊢ 𝑅 = (RingCat‘𝑈)
funcringcsetcALTV2.s	⊢ 𝑆 = (SetCat‘𝑈)
funcringcsetcALTV2.b	⊢ 𝐵 = (Base‘𝑅)
funcringcsetcALTV2.c	⊢ 𝐶 = (Base‘𝑆)
funcringcsetcALTV2.u	⊢ (𝜑 → 𝑈 ∈ WUni)
funcringcsetcALTV2.f	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))
funcringcsetcALTV2.g	⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))

Assertion

Ref	Expression
funcringcsetcALTV2lem4	⊢ (𝜑 → 𝐺 Fn (𝐵 × 𝐵))

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

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

Proof of Theorem funcringcsetcALTV2lem4

Step	Hyp	Ref	Expression
1		eqid 2823	. . 3 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦)))
2		ovex 7191	. . . 4 ⊢ (𝑥 RingHom 𝑦) ∈ V
3		id 22	. . . . 5 ⊢ ((𝑥 RingHom 𝑦) ∈ V → (𝑥 RingHom 𝑦) ∈ V)
4	3	resiexd 6981	. . . 4 ⊢ ((𝑥 RingHom 𝑦) ∈ V → ( I ↾ (𝑥 RingHom 𝑦)) ∈ V)
5	2, 4	ax-mp 5	. . 3 ⊢ ( I ↾ (𝑥 RingHom 𝑦)) ∈ V
6	1, 5	fnmpoi 7770	. 2 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))) Fn (𝐵 × 𝐵)
7		funcringcsetcALTV2.g	. . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))
8	7	fneq1d 6448	. 2 ⊢ (𝜑 → (𝐺 Fn (𝐵 × 𝐵) ↔ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))) Fn (𝐵 × 𝐵)))
9	6, 8	mpbiri 260	1 ⊢ (𝜑 → 𝐺 Fn (𝐵 × 𝐵))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  Vcvv 3496   ↦ cmpt 5148   I cid 5461   × cxp 5555   ↾ cres 5559   Fn wfn 6352  ‘cfv 6357  (class class class)co 7158   ∈ cmpo 7160  WUnicwun 10124  Basecbs 16485  SetCatcsetc 17337   RingHom crh 19466  RingCatcringc 44281

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-1st 7691  df-2nd 7692

This theorem is referenced by:  funcringcsetcALTV2  44323