Theorem cmpcov 23707

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 4877	. . . . 5 ⊢ (𝑟 = 𝑆 → ∪ 𝑟 = ∪ 𝑆)
2	1	eqeq2d 2771	. . . 4 ⊢ (𝑟 = 𝑆 → (𝑋 = ∪ 𝑟 ↔ 𝑋 = ∪ 𝑆))
3		pweq 4570	. . . . . 6 ⊢ (𝑟 = 𝑆 → 𝒫 𝑟 = 𝒫 𝑆)
4	3	ineq1d 4116	. . . . 5 ⊢ (𝑟 = 𝑆 → (𝒫 𝑟 ∩ Fin) = (𝒫 𝑆 ∩ Fin))
5	4	rexeqdv 3320	. . . 4 ⊢ (𝑟 = 𝑆 → (∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = ∪ 𝑠 ↔ ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = ∪ 𝑠))
6	2, 5	imbi12d 347	. . 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 23706	. . . . 5 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑟 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = ∪ 𝑠)))
9	8	simprbi 503	. . . 4 ⊢ (𝐽 ∈ Comp → ∀𝑟 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = ∪ 𝑠))
10	9	adantr 486	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽) → ∀𝑟 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = ∪ 𝑠))
11		ssexg 5280	. . . . 5 ⊢ ((𝑆 ⊆ 𝐽 ∧ 𝐽 ∈ Comp) → 𝑆 ∈ V)
12	11	ancoms 464	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽) → 𝑆 ∈ V)
13		simpr 490	. . . 4 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽) → 𝑆 ⊆ 𝐽)
14	12, 13	elpwd 4562	. . 3 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽) → 𝑆 ∈ 𝒫 𝐽)
15	6, 10, 14	rspcdva 3577	. 2 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽) → (𝑋 = ∪ 𝑆 → ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = ∪ 𝑠))
16	15	3impia 1135	1 ⊢ ((𝐽 ∈ Comp ∧ 𝑆 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑆) → ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = ∪ 𝑠)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3895   ∩ cin 4089  𝒫 cpw 4556  ∪ cuni 4866  Fincfn 8973  Topctop 23211  Compccmp 23704

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-in 4091  df-pw 4558  df-uni 4867  df-cmp 23705

This theorem is used by:  cmpcov2  23708  cncmp  23710  discmp  23716  cmpcld  23720  sscmp  23723  comppfsc  23851  alexsubALTlem1  24366  ptcmplem3  24373  lebnum  25285  heibor1  38787  tmachlem-extpcover  47998