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Theorem funcringcsetclem5ALTV 47491
Description: Lemma 5 for funcringcsetcALTV 47496. (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
funcringcsetclem5ALTV ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (π‘‹πΊπ‘Œ) = ( I β†Ύ (𝑋 RingHom π‘Œ)))
Distinct variable groups:   π‘₯,𝐡   π‘₯,𝑋   πœ‘,π‘₯   π‘₯,𝐢   𝑦,𝐡,π‘₯   𝑦,𝑋   π‘₯,π‘Œ,𝑦   πœ‘,𝑦
Allowed substitution hints:   𝐢(𝑦)   𝑅(π‘₯,𝑦)   𝑆(π‘₯,𝑦)   π‘ˆ(π‘₯,𝑦)   𝐹(π‘₯,𝑦)   𝐺(π‘₯,𝑦)
Proof of Theorem funcringcsetclem5ALTV
StepHypRef Expression
1 funcringcsetcALTV.g . . 3 (πœ‘ β†’ 𝐺 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ (π‘₯ RingHom 𝑦))))
21adantr 479 . 2 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ 𝐺 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ (π‘₯ RingHom 𝑦))))
3 oveq12 7426 . . . 4 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ) β†’ (π‘₯ RingHom 𝑦) = (𝑋 RingHom π‘Œ))
43adantl 480 . . 3 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) ∧ (π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ)) β†’ (π‘₯ RingHom 𝑦) = (𝑋 RingHom π‘Œ))
54reseq2d 5984 . 2 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) ∧ (π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ)) β†’ ( I β†Ύ (π‘₯ RingHom 𝑦)) = ( I β†Ύ (𝑋 RingHom π‘Œ)))
6 simprl 769 . 2 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ 𝑋 ∈ 𝐡)
7 simprr 771 . 2 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ π‘Œ ∈ 𝐡)
8 ovexd 7452 . . 3 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (𝑋 RingHom π‘Œ) ∈ V)
98resiexd 7226 . 2 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ ( I β†Ύ (𝑋 RingHom π‘Œ)) ∈ V)
102, 5, 6, 7, 9ovmpod 7571 1 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (π‘‹πΊπ‘Œ) = ( I β†Ύ (𝑋 RingHom π‘Œ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3463   ↦ cmpt 5231   I cid 5574   β†Ύ cres 5679  β€˜cfv 6547  (class class class)co 7417   ∈ cmpo 7419  WUnicwun 10723  Basecbs 17179  SetCatcsetc 18063   RingHom crh 20412  RingCatALTVcringcALTV 47461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6499  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-ov 7420  df-oprab 7421  df-mpo 7422
This theorem is referenced by:  funcringcsetclem6ALTV  47492  funcringcsetclem7ALTV  47493  funcringcsetclem8ALTV  47494  funcringcsetclem9ALTV  47495
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