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Theorem funcringcsetclem1ALTV 47176
Description: Lemma 1 for funcringcsetcALTV 47185. (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)))
Assertion
Ref Expression
funcringcsetclem1ALTV ((𝜑𝑋𝐵) → (𝐹𝑋) = (Base‘𝑋))
Distinct variable groups:   𝑥,𝐵   𝑥,𝑋   𝜑,𝑥
Allowed substitution hints:   𝐶(𝑥)   𝑅(𝑥)   𝑆(𝑥)   𝑈(𝑥)   𝐹(𝑥)

Proof of Theorem funcringcsetclem1ALTV
StepHypRef Expression
1 funcringcsetcALTV.f . . 3 (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
21adantr 480 . 2 ((𝜑𝑋𝐵) → 𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
3 fveq2 6881 . . 3 (𝑥 = 𝑋 → (Base‘𝑥) = (Base‘𝑋))
43adantl 481 . 2 (((𝜑𝑋𝐵) ∧ 𝑥 = 𝑋) → (Base‘𝑥) = (Base‘𝑋))
5 simpr 484 . 2 ((𝜑𝑋𝐵) → 𝑋𝐵)
6 fvexd 6896 . 2 ((𝜑𝑋𝐵) → (Base‘𝑋) ∈ V)
72, 4, 5, 6fvmptd 6995 1 ((𝜑𝑋𝐵) → (𝐹𝑋) = (Base‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  Vcvv 3466  cmpt 5221  cfv 6533  WUnicwun 10691  Basecbs 17143  SetCatcsetc 18027  RingCatALTVcringcALTV 47150
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 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6485  df-fun 6535  df-fv 6541
This theorem is referenced by:  funcringcsetclem2ALTV  47177  funcringcsetclem7ALTV  47182  funcringcsetclem8ALTV  47183  funcringcsetclem9ALTV  47184
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