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Theorem funcringcsetclem5ALTV 41834
Description: Lemma 5 for funcringcsetcALTV 41839. (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 481 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))
3 oveq12 6656 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑥 RingHom 𝑦) = (𝑋 RingHom 𝑌))
43adantl 482 . . 3 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥 RingHom 𝑦) = (𝑋 RingHom 𝑌))
54reseq2d 5394 . 2 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ( I ↾ (𝑥 RingHom 𝑦)) = ( I ↾ (𝑋 RingHom 𝑌)))
6 simprl 794 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
7 simprr 796 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
8 ovexd 6677 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 RingHom 𝑌) ∈ V)
98resiexd 6477 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ( I ↾ (𝑋 RingHom 𝑌)) ∈ V)
102, 5, 6, 7, 9ovmpt2d 6785 1 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑋 RingHom 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1482  wcel 1989  Vcvv 3198  cmpt 4727   I cid 5021  cres 5114  cfv 5886  (class class class)co 6647  cmpt2 6649  WUnicwun 9519  Basecbs 15851  SetCatcsetc 16719   RingHom crh 18706  RingCatALTVcringcALTV 41775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652
This theorem is referenced by:  funcringcsetclem6ALTV  41835  funcringcsetclem7ALTV  41836  funcringcsetclem8ALTV  41837  funcringcsetclem9ALTV  41838
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