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Theorem funcringcsetcALTV2lem1 44298
Description: Lemma 1 for funcringcsetcALTV2 44307. (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 483 . 2 ((𝜑𝑋𝐵) → 𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
3 fveq2 6663 . . 3 (𝑥 = 𝑋 → (Base‘𝑥) = (Base‘𝑋))
43adantl 484 . 2 (((𝜑𝑋𝐵) ∧ 𝑥 = 𝑋) → (Base‘𝑥) = (Base‘𝑋))
5 simpr 487 . 2 ((𝜑𝑋𝐵) → 𝑋𝐵)
6 fvexd 6678 . 2 ((𝜑𝑋𝐵) → (Base‘𝑋) ∈ V)
72, 4, 5, 6fvmptd 6768 1 ((𝜑𝑋𝐵) → (𝐹𝑋) = (Base‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1531  wcel 2108  Vcvv 3493  cmpt 5137  cfv 6348  WUnicwun 10114  Basecbs 16475  SetCatcsetc 17327  RingCatcringc 44265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356
This theorem is referenced by:  funcringcsetcALTV2lem2  44299  funcringcsetcALTV2lem7  44304  funcringcsetcALTV2lem8  44305  funcringcsetcALTV2lem9  44306
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