Theorem isptfin 23847

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	⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ PtFin ↔ ∀𝑥 ∈ 𝑋 {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦} ∈ 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.

Step	Hyp	Ref	Expression
1		unieq 4880	. . . 4 ⊢ (𝑎 = 𝐴 → ∪ 𝑎 = ∪ 𝐴)
2		isptfin.1	. . . 4 ⊢ 𝑋 = ∪ 𝐴
3	1, 2	eqtr4di 2816	. . 3 ⊢ (𝑎 = 𝐴 → ∪ 𝑎 = 𝑋)
4		rabeq 3429	. . . 4 ⊢ (𝑎 = 𝐴 → {𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} = {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦})
5	4	eleq1d 2848	. . 3 ⊢ (𝑎 = 𝐴 → ({𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} ∈ Fin ↔ {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦} ∈ Fin))
6	3, 5	raleqbidv 3337	. 2 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ ∪ 𝑎{𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} ∈ Fin ↔ ∀𝑥 ∈ 𝑋 {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦} ∈ Fin))
7		df-ptfin 23837	. 2 ⊢ PtFin = {𝑎 ∣ ∀𝑥 ∈ ∪ 𝑎{𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} ∈ Fin}
8	6, 7	elab2g 3636	1 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ PtFin ↔ ∀𝑥 ∈ 𝑋 {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦} ∈ Fin))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1569   ∈ wcel 2146  ∀wral 3079  {crab 3415  ∪ cuni 4869  Fincfn 8984  PtFincptfin 23834

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rab 3416  df-v 3455  df-ss 3900  df-uni 4870  df-ptfin 23837

This theorem is used by:  finptfin  23849  ptfinfin  23850  lfinpfin  23855