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

Theorem funcsetcestrclem1 17002
Description: Lemma 1 for funcsetcestrc 17012. (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 466 . 2 ((𝜑𝑋𝐶) → 𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))
3 opeq2 4541 . . . 4 (𝑥 = 𝑋 → ⟨(Base‘ndx), 𝑥⟩ = ⟨(Base‘ndx), 𝑋⟩)
43sneqd 4329 . . 3 (𝑥 = 𝑋 → {⟨(Base‘ndx), 𝑥⟩} = {⟨(Base‘ndx), 𝑋⟩})
54adantl 467 . 2 (((𝜑𝑋𝐶) ∧ 𝑥 = 𝑋) → {⟨(Base‘ndx), 𝑥⟩} = {⟨(Base‘ndx), 𝑋⟩})
6 simpr 471 . 2 ((𝜑𝑋𝐶) → 𝑋𝐶)
7 snex 5037 . . 3 {⟨(Base‘ndx), 𝑋⟩} ∈ V
87a1i 11 . 2 ((𝜑𝑋𝐶) → {⟨(Base‘ndx), 𝑋⟩} ∈ V)
92, 5, 6, 8fvmptd 6432 1 ((𝜑𝑋𝐶) → (𝐹𝑋) = {⟨(Base‘ndx), 𝑋⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  {csn 4317  cop 4323  cmpt 4864  cfv 6030  ndxcnx 16061  Basecbs 16064  SetCatcsetc 16932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038
This theorem is referenced by:  funcsetcestrclem2  17003  embedsetcestrclem  17005  funcsetcestrclem7  17009  funcsetcestrclem8  17010  funcsetcestrclem9  17011  fullsetcestrc  17014
  Copyright terms: Public domain W3C validator