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Theorem funcsetcestrclem1 16715
Description: Lemma 1 for funcsetcestrc 16725. (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 481 . 2 ((𝜑𝑋𝐶) → 𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))
3 opeq2 4371 . . . 4 (𝑥 = 𝑋 → ⟨(Base‘ndx), 𝑥⟩ = ⟨(Base‘ndx), 𝑋⟩)
43sneqd 4160 . . 3 (𝑥 = 𝑋 → {⟨(Base‘ndx), 𝑥⟩} = {⟨(Base‘ndx), 𝑋⟩})
54adantl 482 . 2 (((𝜑𝑋𝐶) ∧ 𝑥 = 𝑋) → {⟨(Base‘ndx), 𝑥⟩} = {⟨(Base‘ndx), 𝑋⟩})
6 simpr 477 . 2 ((𝜑𝑋𝐶) → 𝑋𝐶)
7 snex 4869 . . 3 {⟨(Base‘ndx), 𝑋⟩} ∈ V
87a1i 11 . 2 ((𝜑𝑋𝐶) → {⟨(Base‘ndx), 𝑋⟩} ∈ V)
92, 5, 6, 8fvmptd 6245 1 ((𝜑𝑋𝐶) → (𝐹𝑋) = {⟨(Base‘ndx), 𝑋⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  {csn 4148  cop 4154  cmpt 4673  cfv 5847  ndxcnx 15778  Basecbs 15781  SetCatcsetc 16646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855
This theorem is referenced by:  funcsetcestrclem2  16716  embedsetcestrclem  16718  funcsetcestrclem7  16722  funcsetcestrclem8  16723  funcsetcestrclem9  16724  fullsetcestrc  16727
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