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

Theorem isptfin 23371
Description: The statement "is a point-finite cover." (Contributed by Jeff Hankins, 21-Jan-2010.)
Hypothesis
Ref Expression
isptfin.1 𝑋 = 𝐴
Assertion
Ref Expression
isptfin (𝐴𝐵 → (𝐴 ∈ PtFin ↔ ∀𝑥𝑋 {𝑦𝐴𝑥𝑦} ∈ Fin))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑋
Allowed substitution hints:   𝐵(𝑥,𝑦)   𝑋(𝑦)

Proof of Theorem isptfin
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 unieq 4913 . . . 4 (𝑎 = 𝐴 𝑎 = 𝐴)
2 isptfin.1 . . . 4 𝑋 = 𝐴
31, 2eqtr4di 2784 . . 3 (𝑎 = 𝐴 𝑎 = 𝑋)
4 rabeq 3440 . . . 4 (𝑎 = 𝐴 → {𝑦𝑎𝑥𝑦} = {𝑦𝐴𝑥𝑦})
54eleq1d 2812 . . 3 (𝑎 = 𝐴 → ({𝑦𝑎𝑥𝑦} ∈ Fin ↔ {𝑦𝐴𝑥𝑦} ∈ Fin))
63, 5raleqbidv 3336 . 2 (𝑎 = 𝐴 → (∀𝑥 𝑎{𝑦𝑎𝑥𝑦} ∈ Fin ↔ ∀𝑥𝑋 {𝑦𝐴𝑥𝑦} ∈ Fin))
7 df-ptfin 23361 . 2 PtFin = {𝑎 ∣ ∀𝑥 𝑎{𝑦𝑎𝑥𝑦} ∈ Fin}
86, 7elab2g 3665 1 (𝐴𝐵 → (𝐴 ∈ PtFin ↔ ∀𝑥𝑋 {𝑦𝐴𝑥𝑦} ∈ Fin))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wcel 2098  wral 3055  {crab 3426   cuni 4902  Fincfn 8938  PtFincptfin 23358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rab 3427  df-v 3470  df-in 3950  df-ss 3960  df-uni 4903  df-ptfin 23361
This theorem is referenced by:  finptfin  23373  ptfinfin  23374  lfinpfin  23379
  Copyright terms: Public domain W3C validator