Theorem ptfinfin 22129

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 22126	. . . 4 ⊢ (𝐴 ∈ PtFin → (𝐴 ∈ PtFin ↔ ∀𝑝 ∈ 𝑋 {𝑥 ∈ 𝐴 ∣ 𝑝 ∈ 𝑥} ∈ Fin))
3	2	ibi 269	. . 3 ⊢ (𝐴 ∈ PtFin → ∀𝑝 ∈ 𝑋 {𝑥 ∈ 𝐴 ∣ 𝑝 ∈ 𝑥} ∈ Fin)
4		eleq1 2902	. . . . . 6 ⊢ (𝑝 = 𝑃 → (𝑝 ∈ 𝑥 ↔ 𝑃 ∈ 𝑥))
5	4	rabbidv 3482	. . . . 5 ⊢ (𝑝 = 𝑃 → {𝑥 ∈ 𝐴 ∣ 𝑝 ∈ 𝑥} = {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥})
6	5	eleq1d 2899	. . . 4 ⊢ (𝑝 = 𝑃 → ({𝑥 ∈ 𝐴 ∣ 𝑝 ∈ 𝑥} ∈ Fin ↔ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥} ∈ Fin))
7	6	rspccv 3622	. . 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 409	1 ⊢ ((𝐴 ∈ PtFin ∧ 𝑃 ∈ 𝑋) → {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥} ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  {crab 3144  ∪ cuni 4840  Fincfn 8511  PtFincptfin 22113

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rab 3149  df-v 3498  df-in 3945  df-ss 3954  df-uni 4841  df-ptfin 22116

This theorem is referenced by:  locfindis  22140  comppfsc  22142