Theorem isptfin 23426

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 4865	. . . 4 ⊢ (𝑎 = 𝐴 → ∪ 𝑎 = ∪ 𝐴)
2		isptfin.1	. . . 4 ⊢ 𝑋 = ∪ 𝐴
3	1, 2	eqtr4di 2784	. . 3 ⊢ (𝑎 = 𝐴 → ∪ 𝑎 = 𝑋)
4		rabeq 3409	. . . 4 ⊢ (𝑎 = 𝐴 → {𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} = {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦})
5	4	eleq1d 2816	. . 3 ⊢ (𝑎 = 𝐴 → ({𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} ∈ Fin ↔ {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦} ∈ Fin))
6	3, 5	raleqbidv 3312	. 2 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ ∪ 𝑎{𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} ∈ Fin ↔ ∀𝑥 ∈ 𝑋 {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦} ∈ Fin))
7		df-ptfin 23416	. 2 ⊢ PtFin = {𝑎 ∣ ∀𝑥 ∈ ∪ 𝑎{𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} ∈ Fin}
8	6, 7	elab2g 3631	1 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ PtFin ↔ ∀𝑥 ∈ 𝑋 {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦} ∈ Fin))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2111  ∀wral 3047  {crab 3395  ∪ cuni 4854  Fincfn 8864  PtFincptfin 23413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rab 3396  df-v 3438  df-ss 3914  df-uni 4855  df-ptfin 23416

This theorem is referenced by:  finptfin  23428  ptfinfin  23429  lfinpfin  23434