Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  funcringcsetcALTV2lem1 Structured version   Visualization version   GIF version

Theorem funcringcsetcALTV2lem1 42830
Description: Lemma 1 for funcringcsetcALTV2 42839. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
Hypotheses
Ref Expression
funcringcsetcALTV2.r 𝑅 = (RingCat‘𝑈)
funcringcsetcALTV2.s 𝑆 = (SetCat‘𝑈)
funcringcsetcALTV2.b 𝐵 = (Base‘𝑅)
funcringcsetcALTV2.c 𝐶 = (Base‘𝑆)
funcringcsetcALTV2.u (𝜑𝑈 ∈ WUni)
funcringcsetcALTV2.f (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
Assertion
Ref Expression
funcringcsetcALTV2lem1 ((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 funcringcsetcALTV2lem1
StepHypRef Expression
1 funcringcsetcALTV2.f . . 3 (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
21adantr 473 . 2 ((𝜑𝑋𝐵) → 𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
3 fveq2 6412 . . 3 (𝑥 = 𝑋 → (Base‘𝑥) = (Base‘𝑋))
43adantl 474 . 2 (((𝜑𝑋𝐵) ∧ 𝑥 = 𝑋) → (Base‘𝑥) = (Base‘𝑋))
5 simpr 478 . 2 ((𝜑𝑋𝐵) → 𝑋𝐵)
6 fvexd 6427 . 2 ((𝜑𝑋𝐵) → (Base‘𝑋) ∈ V)
72, 4, 5, 6fvmptd 6514 1 ((𝜑𝑋𝐵) → (𝐹𝑋) = (Base‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  Vcvv 3386  cmpt 4923  cfv 6102  WUnicwun 9811  Basecbs 16183  SetCatcsetc 17038  RingCatcringc 42797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pr 5098
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-iota 6065  df-fun 6104  df-fv 6110
This theorem is referenced by:  funcringcsetcALTV2lem2  42831  funcringcsetcALTV2lem7  42836  funcringcsetcALTV2lem8  42837  funcringcsetcALTV2lem9  42838
  Copyright terms: Public domain W3C validator