Theorem cmpcov 21605

Description: An open cover of a compact topology has a finite subcover. (Contributed by Jeff Hankins, 29-Jun-2009.)

Hypothesis

Ref	Expression
iscmp.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
cmpcov	⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑆) → ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = ∪ 𝑠)

Distinct variable groups:   𝐽,𝑠   𝑆,𝑠

Allowed substitution hint:   𝑋(𝑠)

Proof of Theorem cmpcov

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unieq 4681	. . . . 5 ⊢ (𝑟 = 𝑆 → ∪ 𝑟 = ∪ 𝑆)
2	1	eqeq2d 2788	. . . 4 ⊢ (𝑟 = 𝑆 → (𝑋 = ∪ 𝑟 ↔ 𝑋 = ∪ 𝑆))
3		pweq 4382	. . . . . 6 ⊢ (𝑟 = 𝑆 → 𝒫 𝑟 = 𝒫 𝑆)
4	3	ineq1d 4036	. . . . 5 ⊢ (𝑟 = 𝑆 → (𝒫 𝑟 ∩ Fin) = (𝒫 𝑆 ∩ Fin))
5	4	rexeqdv 3341	. . . 4 ⊢ (𝑟 = 𝑆 → (∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = ∪ 𝑠 ↔ ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = ∪ 𝑠))
6	2, 5	imbi12d 336	. . 3 ⊢ (𝑟 = 𝑆 → ((𝑋 = ∪ 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = ∪ 𝑠) ↔ (𝑋 = ∪ 𝑆 → ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = ∪ 𝑠)))
7		iscmp.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
8	7	iscmp 21604	. . . . 5 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑟 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = ∪ 𝑠)))
9	8	simprbi 492	. . . 4 ⊢ (𝐽 ∈ Comp → ∀𝑟 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = ∪ 𝑠))
10	9	adantr 474	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽) → ∀𝑟 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = ∪ 𝑠))
11		simpr 479	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽) → 𝑆 ⊆ 𝐽)
12		ssexg 5043	. . . . . 6 ⊢ ((𝑆 ⊆ 𝐽 ∧ 𝐽 ∈ Comp) → 𝑆 ∈ V)
13	12	ancoms 452	. . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽) → 𝑆 ∈ V)
14		elpwg 4387	. . . . 5 ⊢ (𝑆 ∈ V → (𝑆 ∈ 𝒫 𝐽 ↔ 𝑆 ⊆ 𝐽))
15	13, 14	syl 17	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽) → (𝑆 ∈ 𝒫 𝐽 ↔ 𝑆 ⊆ 𝐽))
16	11, 15	mpbird 249	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽) → 𝑆 ∈ 𝒫 𝐽)
17	6, 10, 16	rspcdva 3517	. 2 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽) → (𝑋 = ∪ 𝑆 → ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = ∪ 𝑠))
18	17	3impia 1106	1 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑆) → ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = ∪ 𝑠)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2107  ∀wral 3090  ∃wrex 3091  Vcvv 3398   ∩ cin 3791   ⊆ wss 3792  𝒫 cpw 4379  ∪ cuni 4673  Fincfn 8243  Topctop 21109  Compccmp 21602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754  ax-sep 5019

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-in 3799  df-ss 3806  df-pw 4381  df-uni 4674  df-cmp 21603

This theorem is referenced by:  cmpcov2  21606  cncmp  21608  discmp  21614  cmpcld  21618  sscmp  21621  comppfsc  21748  alexsubALTlem1  22263  ptcmplem3  22270  lebnum  23175  heibor1  34238