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Theorem funcringcsetclem5ALTV 47154
Description: Lemma 5 for funcringcsetcALTV 47159. (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
funcringcsetclem5ALTV ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑋 RingHom 𝑌)))
Distinct variable groups:   𝑥,𝐵   𝑥,𝑋   𝜑,𝑥   𝑥,𝐶   𝑦,𝐵,𝑥   𝑦,𝑋   𝑥,𝑌,𝑦   𝜑,𝑦
Allowed substitution hints:   𝐶(𝑦)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem funcringcsetclem5ALTV
StepHypRef Expression
1 funcringcsetcALTV.g . . 3 (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))
21adantr 480 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))
3 oveq12 7421 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥 RingHom 𝑦) = (𝑋 RingHom 𝑌))
43adantl 481 . . 3 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥 RingHom 𝑦) = (𝑋 RingHom 𝑌))
54reseq2d 5981 . 2 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ( I ↾ (𝑥 RingHom 𝑦)) = ( I ↾ (𝑋 RingHom 𝑌)))
6 simprl 768 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
7 simprr 770 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
8 ovexd 7447 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 RingHom 𝑌) ∈ V)
98resiexd 7220 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ( I ↾ (𝑋 RingHom 𝑌)) ∈ V)
102, 5, 6, 7, 9ovmpod 7563 1 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑋 RingHom 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  Vcvv 3473  cmpt 5231   I cid 5573  cres 5678  cfv 6543  (class class class)co 7412  cmpo 7414  WUnicwun 10701  Basecbs 17151  SetCatcsetc 18035   RingHom crh 20367  RingCatALTVcringcALTV 47124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  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 7415  df-oprab 7416  df-mpo 7417
This theorem is referenced by:  funcringcsetclem6ALTV  47155  funcringcsetclem7ALTV  47156  funcringcsetclem8ALTV  47157  funcringcsetclem9ALTV  47158
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