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Theorem funcringcsetcALTV2lem4 41343
 Description: Lemma 4 for funcringcsetcALTV2 41349. (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 (𝜑𝑈 ∈ WUni)
funcringcsetcALTV2.f (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
funcringcsetcALTV2.g (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))
Assertion
Ref Expression
funcringcsetcALTV2lem4 (𝜑𝐺 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
StepHypRef Expression
1 eqid 2621 . . 3 (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))) = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦)))
2 ovex 6635 . . . 4 (𝑥 RingHom 𝑦) ∈ V
3 id 22 . . . . 5 ((𝑥 RingHom 𝑦) ∈ V → (𝑥 RingHom 𝑦) ∈ V)
43resiexd 6437 . . . 4 ((𝑥 RingHom 𝑦) ∈ V → ( I ↾ (𝑥 RingHom 𝑦)) ∈ V)
52, 4ax-mp 5 . . 3 ( I ↾ (𝑥 RingHom 𝑦)) ∈ V
61, 5fnmpt2i 7187 . 2 (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))) Fn (𝐵 × 𝐵)
7 funcringcsetcALTV2.g . . 3 (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))
87fneq1d 5941 . 2 (𝜑 → (𝐺 Fn (𝐵 × 𝐵) ↔ (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))) Fn (𝐵 × 𝐵)))
96, 8mpbiri 248 1 (𝜑𝐺 Fn (𝐵 × 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1480   ∈ wcel 1987  Vcvv 3186   ↦ cmpt 4675   I cid 4986   × cxp 5074   ↾ cres 5078   Fn wfn 5844  ‘cfv 5849  (class class class)co 6607   ↦ cmpt2 6609  WUnicwun 9469  Basecbs 15784  SetCatcsetc 16649   RingHom crh 18636  RingCatcringc 41307 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-1st 7116  df-2nd 7117 This theorem is referenced by:  funcringcsetcALTV2  41349
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