Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  funcringcsetclem5ALTV Structured version   Visualization version   GIF version

Theorem funcringcsetclem5ALTV 41855
 Description: Lemma 5 for funcringcsetcALTV 41860. (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 6558 . . . 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 5317 . 2 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ( I ↾ (𝑥 RingHom 𝑦)) = ( I ↾ (𝑋 RingHom 𝑌)))
6 simprl 790 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
7 simprr 792 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
8 ovex 6577 . . . 4 (𝑋 RingHom 𝑌) ∈ V
98a1i 11 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 RingHom 𝑌) ∈ V)
109resiexd 6385 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ( I ↾ (𝑋 RingHom 𝑌)) ∈ V)
112, 5, 6, 7, 10ovmpt2d 6686 1 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑋 RingHom 𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ↦ cmpt 4643   I cid 4948   ↾ cres 5040  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  WUnicwun 9401  Basecbs 15695  SetCatcsetc 16548   RingHom crh 18535  RingCatALTVcringcALTV 41796 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554 This theorem is referenced by:  funcringcsetclem6ALTV  41856  funcringcsetclem7ALTV  41857  funcringcsetclem8ALTV  41858  funcringcsetclem9ALTV  41859
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