Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  funcsetcestrclem1 Structured version   Visualization version   GIF version

Theorem funcsetcestrclem1 17471
 Description: Lemma 1 for funcsetcestrc 17481. (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 485 . 2 ((𝜑𝑋𝐶) → 𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))
3 opeq2 4764 . . . 4 (𝑥 = 𝑋 → ⟨(Base‘ndx), 𝑥⟩ = ⟨(Base‘ndx), 𝑋⟩)
43sneqd 4535 . . 3 (𝑥 = 𝑋 → {⟨(Base‘ndx), 𝑥⟩} = {⟨(Base‘ndx), 𝑋⟩})
54adantl 486 . 2 (((𝜑𝑋𝐶) ∧ 𝑥 = 𝑋) → {⟨(Base‘ndx), 𝑥⟩} = {⟨(Base‘ndx), 𝑋⟩})
6 simpr 489 . 2 ((𝜑𝑋𝐶) → 𝑋𝐶)
7 snex 5301 . . 3 {⟨(Base‘ndx), 𝑋⟩} ∈ V
87a1i 11 . 2 ((𝜑𝑋𝐶) → {⟨(Base‘ndx), 𝑋⟩} ∈ V)
92, 5, 6, 8fvmptd 6767 1 ((𝜑𝑋𝐶) → (𝐹𝑋) = {⟨(Base‘ndx), 𝑋⟩})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   = wceq 1539   ∈ wcel 2112  Vcvv 3410  {csn 4523  ⟨cop 4529   ↦ cmpt 5113  ‘cfv 6336  ndxcnx 16539  Basecbs 16542  SetCatcsetc 17402 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-iota 6295  df-fun 6338  df-fv 6344 This theorem is referenced by:  funcsetcestrclem2  17472  embedsetcestrclem  17474  funcsetcestrclem7  17478  funcsetcestrclem8  17479  funcsetcestrclem9  17480  fullsetcestrc  17483
 Copyright terms: Public domain W3C validator