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Theorem funcringcsetclem4ALTV 47301
Description: Lemma 4 for funcringcsetcALTV 47307. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
Hypotheses
Ref Expression
funcringcsetcALTV.r 𝑅 = (RingCatALTVβ€˜π‘ˆ)
funcringcsetcALTV.s 𝑆 = (SetCatβ€˜π‘ˆ)
funcringcsetcALTV.b 𝐡 = (Baseβ€˜π‘…)
funcringcsetcALTV.c 𝐢 = (Baseβ€˜π‘†)
funcringcsetcALTV.u (πœ‘ β†’ π‘ˆ ∈ WUni)
funcringcsetcALTV.f (πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝐡 ↦ (Baseβ€˜π‘₯)))
funcringcsetcALTV.g (πœ‘ β†’ 𝐺 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ (π‘₯ RingHom 𝑦))))
Assertion
Ref Expression
funcringcsetclem4ALTV (πœ‘ β†’ 𝐺 Fn (𝐡 Γ— 𝐡))
Distinct variable groups:   π‘₯,𝐡   πœ‘,π‘₯   π‘₯,𝐢   𝑦,𝐡,π‘₯
Allowed substitution hints:   πœ‘(𝑦)   𝐢(𝑦)   𝑅(π‘₯,𝑦)   𝑆(π‘₯,𝑦)   π‘ˆ(π‘₯,𝑦)   𝐹(π‘₯,𝑦)   𝐺(π‘₯,𝑦)

Proof of Theorem funcringcsetclem4ALTV
StepHypRef Expression
1 eqid 2727 . . 3 (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ (π‘₯ RingHom 𝑦))) = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ (π‘₯ RingHom 𝑦)))
2 ovex 7447 . . . 4 (π‘₯ RingHom 𝑦) ∈ V
3 id 22 . . . . 5 ((π‘₯ RingHom 𝑦) ∈ V β†’ (π‘₯ RingHom 𝑦) ∈ V)
43resiexd 7222 . . . 4 ((π‘₯ RingHom 𝑦) ∈ V β†’ ( I β†Ύ (π‘₯ RingHom 𝑦)) ∈ V)
52, 4ax-mp 5 . . 3 ( I β†Ύ (π‘₯ RingHom 𝑦)) ∈ V
61, 5fnmpoi 8068 . 2 (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ (π‘₯ RingHom 𝑦))) Fn (𝐡 Γ— 𝐡)
7 funcringcsetcALTV.g . . 3 (πœ‘ β†’ 𝐺 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ (π‘₯ RingHom 𝑦))))
87fneq1d 6641 . 2 (πœ‘ β†’ (𝐺 Fn (𝐡 Γ— 𝐡) ↔ (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ (π‘₯ RingHom 𝑦))) Fn (𝐡 Γ— 𝐡)))
96, 8mpbiri 258 1 (πœ‘ β†’ 𝐺 Fn (𝐡 Γ— 𝐡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1534   ∈ wcel 2099  Vcvv 3469   ↦ cmpt 5225   I cid 5569   Γ— cxp 5670   β†Ύ cres 5674   Fn wfn 6537  β€˜cfv 6542  (class class class)co 7414   ∈ cmpo 7416  WUnicwun 10715  Basecbs 17171  SetCatcsetc 18055   RingHom crh 20397  RingCatALTVcringcALTV 47272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988
This theorem is referenced by:  funcringcsetcALTV  47307
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