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Theorem funcestrcsetclem1 16544
Description: Lemma 1 for funcestrcsetc 16553. (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 479 . 2 ((𝜑𝑋𝐵) → 𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
3 fveq2 6083 . . 3 (𝑥 = 𝑋 → (Base‘𝑥) = (Base‘𝑋))
43adantl 480 . 2 (((𝜑𝑋𝐵) ∧ 𝑥 = 𝑋) → (Base‘𝑥) = (Base‘𝑋))
5 simpr 475 . 2 ((𝜑𝑋𝐵) → 𝑋𝐵)
6 fvex 6093 . . 3 (Base‘𝑋) ∈ V
76a1i 11 . 2 ((𝜑𝑋𝐵) → (Base‘𝑋) ∈ V)
82, 4, 5, 7fvmptd 6177 1 ((𝜑𝑋𝐵) → (𝐹𝑋) = (Base‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1975  Vcvv 3167  cmpt 4632  cfv 5785  WUnicwun 9373  Basecbs 15636  SetCatcsetc 16489  ExtStrCatcestrc 16526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-sep 4698  ax-nul 4707  ax-pr 4823
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ral 2895  df-rex 2896  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-br 4573  df-opab 4633  df-mpt 4634  df-id 4938  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-iota 5749  df-fun 5787  df-fv 5793
This theorem is referenced by:  funcestrcsetclem2  16545  funcestrcsetclem7  16550  funcestrcsetclem8  16551  funcestrcsetclem9  16552  fullestrcsetc  16555  equivestrcsetc  16556
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