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Theorem funcringcsetclem4ALTV 46434
Description: Lemma 4 for funcringcsetcALTV 46440. (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 2737 . . 3 (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ (π‘₯ RingHom 𝑦))) = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ (π‘₯ RingHom 𝑦)))
2 ovex 7395 . . . 4 (π‘₯ RingHom 𝑦) ∈ V
3 id 22 . . . . 5 ((π‘₯ RingHom 𝑦) ∈ V β†’ (π‘₯ RingHom 𝑦) ∈ V)
43resiexd 7171 . . . 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 8007 . 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 6600 . 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 1542   ∈ wcel 2107  Vcvv 3448   ↦ cmpt 5193   I cid 5535   Γ— cxp 5636   β†Ύ cres 5640   Fn wfn 6496  β€˜cfv 6501  (class class class)co 7362   ∈ cmpo 7364  WUnicwun 10643  Basecbs 17090  SetCatcsetc 17968   RingHom crh 20152  RingCatALTVcringcALTV 46376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927
This theorem is referenced by:  funcringcsetcALTV  46440
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