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Theorem isptfin 21818
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 4714 . . . 4 (𝑎 = 𝐴 𝑎 = 𝐴)
2 isptfin.1 . . . 4 𝑋 = 𝐴
31, 2syl6eqr 2826 . . 3 (𝑎 = 𝐴 𝑎 = 𝑋)
4 rabeq 3400 . . . 4 (𝑎 = 𝐴 → {𝑦𝑎𝑥𝑦} = {𝑦𝐴𝑥𝑦})
54eleq1d 2844 . . 3 (𝑎 = 𝐴 → ({𝑦𝑎𝑥𝑦} ∈ Fin ↔ {𝑦𝐴𝑥𝑦} ∈ Fin))
63, 5raleqbidv 3335 . 2 (𝑎 = 𝐴 → (∀𝑥 𝑎{𝑦𝑎𝑥𝑦} ∈ Fin ↔ ∀𝑥𝑋 {𝑦𝐴𝑥𝑦} ∈ Fin))
7 df-ptfin 21808 . 2 PtFin = {𝑎 ∣ ∀𝑥 𝑎{𝑦𝑎𝑥𝑦} ∈ Fin}
86, 7elab2g 3578 1 (𝐴𝐵 → (𝐴 ∈ PtFin ↔ ∀𝑥𝑋 {𝑦𝐴𝑥𝑦} ∈ Fin))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1507  wcel 2048  wral 3082  {crab 3086   cuni 4706  Fincfn 8298  PtFincptfin 21805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-ext 2745
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-uni 4707  df-ptfin 21808
This theorem is referenced by:  finptfin  21820  ptfinfin  21821  lfinpfin  21826
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