Theorem isptfin 22127

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 4852	. . . 4 ⊢ (𝑎 = 𝐴 → ∪ 𝑎 = ∪ 𝐴)
2		isptfin.1	. . . 4 ⊢ 𝑋 = ∪ 𝐴
3	1, 2	syl6eqr 2877	. . 3 ⊢ (𝑎 = 𝐴 → ∪ 𝑎 = 𝑋)
4		rabeq 3486	. . . 4 ⊢ (𝑎 = 𝐴 → {𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} = {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦})
5	4	eleq1d 2900	. . 3 ⊢ (𝑎 = 𝐴 → ({𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} ∈ Fin ↔ {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦} ∈ Fin))
6	3, 5	raleqbidv 3404	. 2 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ ∪ 𝑎{𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} ∈ Fin ↔ ∀𝑥 ∈ 𝑋 {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦} ∈ Fin))
7		df-ptfin 22117	. 2 ⊢ PtFin = {𝑎 ∣ ∀𝑥 ∈ ∪ 𝑎{𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} ∈ Fin}
8	6, 7	elab2g 3671	1 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ PtFin ↔ ∀𝑥 ∈ 𝑋 {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦} ∈ Fin))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1536   ∈ wcel 2113  ∀wral 3141  {crab 3145  ∪ cuni 4841  Fincfn 8512  PtFincptfin 22114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rab 3150  df-v 3499  df-in 3946  df-ss 3955  df-uni 4842  df-ptfin 22117

This theorem is referenced by:  finptfin  22129  ptfinfin  22130  lfinpfin  22135