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Theorem funcestrcsetclem1 18196
Description: Lemma 1 for funcestrcsetc 18205. (Contributed by AV, 22-Mar-2020.)
Hypotheses
Ref Expression
funcestrcsetc.e 𝐸 = (ExtStrCat‘𝑈)
funcestrcsetc.s 𝑆 = (SetCat‘𝑈)
funcestrcsetc.b 𝐵 = (Base‘𝐸)
funcestrcsetc.c 𝐶 = (Base‘𝑆)
funcestrcsetc.u (𝜑𝑈 ∈ WUni)
funcestrcsetc.f (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
Assertion
Ref Expression
funcestrcsetclem1 ((𝜑𝑋𝐵) → (𝐹𝑋) = (Base‘𝑋))
Distinct variable groups:   𝑥,𝐵   𝑥,𝑋   𝜑,𝑥
Allowed substitution hints:   𝐶(𝑥)   𝑆(𝑥)   𝑈(𝑥)   𝐸(𝑥)   𝐹(𝑥)

Proof of Theorem funcestrcsetclem1
StepHypRef Expression
1 funcestrcsetc.f . . 3 (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
21adantr 480 . 2 ((𝜑𝑋𝐵) → 𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
3 fveq2 6907 . . 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 6922 . 2 ((𝜑𝑋𝐵) → (Base‘𝑋) ∈ V)
72, 4, 5, 6fvmptd 7023 1 ((𝜑𝑋𝐵) → (𝐹𝑋) = (Base‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  cmpt 5231  cfv 6563  WUnicwun 10738  Basecbs 17245  SetCatcsetc 18129  ExtStrCatcestrc 18177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571
This theorem is referenced by:  funcestrcsetclem2  18197  funcestrcsetclem7  18202  funcestrcsetclem8  18203  funcestrcsetclem9  18204  fullestrcsetc  18207  equivestrcsetc  18208
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