Theorem isptfin 21319

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 4444	. . . 4 ⊢ (𝑎 = 𝐴 → ∪ 𝑎 = ∪ 𝐴)
2		isptfin.1	. . . 4 ⊢ 𝑋 = ∪ 𝐴
3	1, 2	syl6eqr 2674	. . 3 ⊢ (𝑎 = 𝐴 → ∪ 𝑎 = 𝑋)
4		rabeq 3192	. . . 4 ⊢ (𝑎 = 𝐴 → {𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} = {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦})
5	4	eleq1d 2686	. . 3 ⊢ (𝑎 = 𝐴 → ({𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} ∈ Fin ↔ {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦} ∈ Fin))
6	3, 5	raleqbidv 3152	. 2 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ ∪ 𝑎{𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} ∈ Fin ↔ ∀𝑥 ∈ 𝑋 {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦} ∈ Fin))
7		df-ptfin 21309	. 2 ⊢ PtFin = {𝑎 ∣ ∀𝑥 ∈ ∪ 𝑎{𝑦 ∈ 𝑎 ∣ 𝑥 ∈ 𝑦} ∈ Fin}
8	6, 7	elab2g 3353	1 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ PtFin ↔ ∀𝑥 ∈ 𝑋 {𝑦 ∈ 𝐴 ∣ 𝑥 ∈ 𝑦} ∈ Fin))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916  ∪ cuni 4436  Fincfn 7955  PtFincptfin 21306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-uni 4437  df-ptfin 21309

This theorem is referenced by:  finptfin  21321  ptfinfin  21322  lfinpfin  21327