Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  funcringcsetclem4ALTV Structured version   Visualization version   GIF version

Theorem funcringcsetclem4ALTV 41304
Description: Lemma 4 for funcringcsetcALTV 41310. (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 (𝜑𝑈 ∈ WUni)
funcringcsetcALTV.f (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
funcringcsetcALTV.g (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))
Assertion
Ref Expression
funcringcsetclem4ALTV (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 funcringcsetclem4ALTV
StepHypRef Expression
1 eqid 2626 . . 3 (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))) = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦)))
2 ovex 6633 . . . 4 (𝑥 RingHom 𝑦) ∈ V
3 id 22 . . . . 5 ((𝑥 RingHom 𝑦) ∈ V → (𝑥 RingHom 𝑦) ∈ V)
43resiexd 6435 . . . 4 ((𝑥 RingHom 𝑦) ∈ V → ( I ↾ (𝑥 RingHom 𝑦)) ∈ V)
52, 4ax-mp 5 . . 3 ( I ↾ (𝑥 RingHom 𝑦)) ∈ V
61, 5fnmpt2i 7185 . 2 (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))) Fn (𝐵 × 𝐵)
7 funcringcsetcALTV.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 1992  Vcvv 3191  cmpt 4678   I cid 4989   × cxp 5077  cres 5081   Fn wfn 5845  cfv 5850  (class class class)co 6605  cmpt2 6607  WUnicwun 9467  Basecbs 15776  SetCatcsetc 16641   RingHom crh 18628  RingCatALTVcringcALTV 41246
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117
This theorem is referenced by:  funcringcsetcALTV  41310
  Copyright terms: Public domain W3C validator