Theorem funcringcsetclem4ALTV 44418

Description: Lemma 4 for funcringcsetcALTV 44424. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
funcringcsetclem4ALTV	⊢ (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 funcringcsetclem4ALTV

Step	Hyp	Ref	Expression
1		eqid 2821	. . 3 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦)))
2		ovex 7175	. . . 4 ⊢ (𝑥 RingHom 𝑦) ∈ V
3		id 22	. . . . 5 ⊢ ((𝑥 RingHom 𝑦) ∈ V → (𝑥 RingHom 𝑦) ∈ V)
4	3	resiexd 6965	. . . 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 7754	. 2 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))) Fn (𝐵 × 𝐵)
7		funcringcsetcALTV.g	. . 3 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))
8	7	fneq1d 6432	. 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 3486   ↦ cmpt 5132   I cid 5445   × cxp 5539   ↾ cres 5543   Fn wfn 6336  ‘cfv 6341  (class class class)co 7142   ∈ cmpo 7144  WUnicwun 10108  Basecbs 16466  SetCatcsetc 17318   RingHom crh 19447  RingCatALTVcringcALTV 44360

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 2793  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5252  ax-pr 5316  ax-un 7447

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 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 3488  df-sbc 3764  df-csb 3872  df-dif 3927  df-un 3929  df-in 3931  df-ss 3940  df-nul 4280  df-if 4454  df-sn 4554  df-pr 4556  df-op 4560  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5446  df-xp 5547  df-rel 5548  df-cnv 5549  df-co 5550  df-dm 5551  df-rn 5552  df-res 5553  df-ima 5554  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-f1o 6348  df-fv 6349  df-ov 7145  df-oprab 7146  df-mpo 7147  df-1st 7675  df-2nd 7676

This theorem is referenced by:  funcringcsetcALTV  44424