Theorem ptfinfin 23451

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 23448	. . . 4 ⊢ (𝐴 ∈ PtFin → (𝐴 ∈ PtFin ↔ ∀𝑝 ∈ 𝑋 {𝑥 ∈ 𝐴 ∣ 𝑝 ∈ 𝑥} ∈ Fin))
3	2	ibi 266	. . 3 ⊢ (𝐴 ∈ PtFin → ∀𝑝 ∈ 𝑋 {𝑥 ∈ 𝐴 ∣ 𝑝 ∈ 𝑥} ∈ Fin)
4		eleq1 2817	. . . . . 6 ⊢ (𝑝 = 𝑃 → (𝑝 ∈ 𝑥 ↔ 𝑃 ∈ 𝑥))
5	4	rabbidv 3438	. . . . 5 ⊢ (𝑝 = 𝑃 → {𝑥 ∈ 𝐴 ∣ 𝑝 ∈ 𝑥} = {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥})
6	5	eleq1d 2814	. . . 4 ⊢ (𝑝 = 𝑃 → ({𝑥 ∈ 𝐴 ∣ 𝑝 ∈ 𝑥} ∈ Fin ↔ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥} ∈ Fin))
7	6	rspccv 3608	. . 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 405	1 ⊢ ((𝐴 ∈ PtFin ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥} ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∀wral 3058  {crab 3430  ∪ cuni 4912  Fincfn 8972  PtFincptfin 23435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rab 3431  df-v 3475  df-in 3956  df-ss 3966  df-uni 4913  df-ptfin 23438

This theorem is referenced by:  locfindis  23462  comppfsc  23464