Theorem ptfinfin 21815

Description: A point covered by a point-finite cover is only covered by finitely many elements. (Contributed by Jeff Hankins, 21-Jan-2010.)

Hypothesis

Ref	Expression
ptfinfin.1	⊢ 𝑋 = ∪ 𝐴

Assertion

Ref	Expression
ptfinfin	⊢ ((𝐴 ∈ PtFin ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥} ∈ Fin)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑃   𝑥,𝑋

Proof of Theorem ptfinfin

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ptfinfin.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐴
2	1	isptfin 21812	. . . 4 ⊢ (𝐴 ∈ PtFin → (𝐴 ∈ PtFin ↔ ∀𝑝 ∈ 𝑋 {𝑥 ∈ 𝐴 ∣ 𝑝 ∈ 𝑥} ∈ Fin))
3	2	ibi 268	. . 3 ⊢ (𝐴 ∈ PtFin → ∀𝑝 ∈ 𝑋 {𝑥 ∈ 𝐴 ∣ 𝑝 ∈ 𝑥} ∈ Fin)
4		eleq1 2872	. . . . . 6 ⊢ (𝑝 = 𝑃 → (𝑝 ∈ 𝑥 ↔ 𝑃 ∈ 𝑥))
5	4	rabbidv 3428	. . . . 5 ⊢ (𝑝 = 𝑃 → {𝑥 ∈ 𝐴 ∣ 𝑝 ∈ 𝑥} = {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥})
6	5	eleq1d 2869	. . . 4 ⊢ (𝑝 = 𝑃 → ({𝑥 ∈ 𝐴 ∣ 𝑝 ∈ 𝑥} ∈ Fin ↔ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥} ∈ Fin))
7	6	rspccv 3558	. . 3 ⊢ (∀𝑝 ∈ 𝑋 {𝑥 ∈ 𝐴 ∣ 𝑝 ∈ 𝑥} ∈ Fin → (𝑃 ∈ 𝑋 → {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥} ∈ Fin))
8	3, 7	syl 17	. 2 ⊢ (𝐴 ∈ PtFin → (𝑃 ∈ 𝑋 → {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥} ∈ Fin))
9	8	imp 407	1 ⊢ ((𝐴 ∈ PtFin ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥} ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1525   ∈ wcel 2083  ∀wral 3107  {crab 3111  ∪ cuni 4751  Fincfn 8364  PtFincptfin 21799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-uni 4752  df-ptfin 21802

This theorem is referenced by:  locfindis  21826  comppfsc  21828