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Theorem funcsetcestrclem1 16617
 Description: Lemma 1 for funcsetcestrc 16627. (Contributed by AV, 27-Mar-2020.)
Hypotheses
Ref Expression
funcsetcestrc.s 𝑆 = (SetCat‘𝑈)
funcsetcestrc.c 𝐶 = (Base‘𝑆)
funcsetcestrc.f (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))
Assertion
Ref Expression
funcsetcestrclem1 ((𝜑𝑋𝐶) → (𝐹𝑋) = {⟨(Base‘ndx), 𝑋⟩})
Distinct variable groups:   𝑥,𝐶   𝑥,𝑋   𝜑,𝑥
Allowed substitution hints:   𝑆(𝑥)   𝑈(𝑥)   𝐹(𝑥)

Proof of Theorem funcsetcestrclem1
StepHypRef Expression
1 funcsetcestrc.f . . 3 (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))
21adantr 480 . 2 ((𝜑𝑋𝐶) → 𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))
3 opeq2 4341 . . . 4 (𝑥 = 𝑋 → ⟨(Base‘ndx), 𝑥⟩ = ⟨(Base‘ndx), 𝑋⟩)
43sneqd 4137 . . 3 (𝑥 = 𝑋 → {⟨(Base‘ndx), 𝑥⟩} = {⟨(Base‘ndx), 𝑋⟩})
54adantl 481 . 2 (((𝜑𝑋𝐶) ∧ 𝑥 = 𝑋) → {⟨(Base‘ndx), 𝑥⟩} = {⟨(Base‘ndx), 𝑋⟩})
6 simpr 476 . 2 ((𝜑𝑋𝐶) → 𝑋𝐶)
7 snex 4835 . . 3 {⟨(Base‘ndx), 𝑋⟩} ∈ V
87a1i 11 . 2 ((𝜑𝑋𝐶) → {⟨(Base‘ndx), 𝑋⟩} ∈ V)
92, 5, 6, 8fvmptd 6197 1 ((𝜑𝑋𝐶) → (𝐹𝑋) = {⟨(Base‘ndx), 𝑋⟩})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125  ⟨cop 4131   ↦ cmpt 4643  ‘cfv 5804  ndxcnx 15692  Basecbs 15695  SetCatcsetc 16548 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812 This theorem is referenced by:  funcsetcestrclem2  16618  embedsetcestrclem  16620  funcsetcestrclem7  16624  funcsetcestrclem8  16625  funcsetcestrclem9  16626  fullsetcestrc  16629
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