Theorem isptfin 23545

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 4942	. . . 4 ⊢ (𝑎 = 𝐴 → ∪ 𝑎 = ∪ 𝐴)
2		isptfin.1	. . . 4 ⊢ 𝑋 = ∪ 𝐴
3	1, 2	eqtr4di 2798	. . 3 ⊢ (𝑎 = 𝐴 → ∪ 𝑎 = 𝑋)
4		rabeq 3458	. . . 4 ⊢ (𝑎 = 𝐴 → {𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} = {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦})
5	4	eleq1d 2829	. . 3 ⊢ (𝑎 = 𝐴 → ({𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} ∈ Fin ↔ {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦} ∈ Fin))
6	3, 5	raleqbidv 3354	. 2 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ ∪ 𝑎{𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} ∈ Fin ↔ ∀𝑥 ∈ 𝑋 {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦} ∈ Fin))
7		df-ptfin 23535	. 2 ⊢ PtFin = {𝑎 ∣ ∀𝑥 ∈ ∪ 𝑎{𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} ∈ Fin}
8	6, 7	elab2g 3696	1 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ PtFin ↔ ∀𝑥 ∈ 𝑋 {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦} ∈ Fin))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2108  ∀wral 3067  {crab 3443  ∪ cuni 4931  Fincfn 9003  PtFincptfin 23532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rab 3444  df-v 3490  df-ss 3993  df-uni 4932  df-ptfin 23535

This theorem is referenced by:  finptfin  23547  ptfinfin  23548  lfinpfin  23553