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Theorem funcringcsetcALTV2lem5 46928
Description: Lemma 5 for funcringcsetcALTV2 46933. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
Hypotheses
Ref Expression
funcringcsetcALTV2.r 𝑅 = (RingCatβ€˜π‘ˆ)
funcringcsetcALTV2.s 𝑆 = (SetCatβ€˜π‘ˆ)
funcringcsetcALTV2.b 𝐡 = (Baseβ€˜π‘…)
funcringcsetcALTV2.c 𝐢 = (Baseβ€˜π‘†)
funcringcsetcALTV2.u (πœ‘ β†’ π‘ˆ ∈ WUni)
funcringcsetcALTV2.f (πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝐡 ↦ (Baseβ€˜π‘₯)))
funcringcsetcALTV2.g (πœ‘ β†’ 𝐺 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ (π‘₯ RingHom 𝑦))))
Assertion
Ref Expression
funcringcsetcALTV2lem5 ((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 funcringcsetcALTV2lem5
StepHypRef Expression
1 funcringcsetcALTV2.g . . 3 (πœ‘ β†’ 𝐺 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ (π‘₯ RingHom 𝑦))))
21adantr 481 . 2 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ 𝐺 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ (π‘₯ RingHom 𝑦))))
3 oveq12 7417 . . . 4 ((π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ) β†’ (π‘₯ RingHom 𝑦) = (𝑋 RingHom π‘Œ))
43adantl 482 . . 3 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) ∧ (π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ)) β†’ (π‘₯ RingHom 𝑦) = (𝑋 RingHom π‘Œ))
54reseq2d 5981 . 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 7443 . . 3 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (𝑋 RingHom π‘Œ) ∈ V)
98resiexd 7217 . 2 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ ( I β†Ύ (𝑋 RingHom π‘Œ)) ∈ V)
102, 5, 6, 7, 9ovmpod 7559 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 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474   ↦ cmpt 5231   I cid 5573   β†Ύ cres 5678  β€˜cfv 6543  (class class class)co 7408   ∈ cmpo 7410  WUnicwun 10694  Basecbs 17143  SetCatcsetc 18024   RingHom crh 20247  RingCatcringc 46891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413
This theorem is referenced by:  funcringcsetcALTV2lem6  46929  funcringcsetcALTV2lem7  46930  funcringcsetcALTV2lem8  46931  funcringcsetcALTV2lem9  46932
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