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Theorem cmpcov 23397
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 ∧ 𝑆𝐽𝑋 = 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.
StepHypRef Expression
1 unieq 4918 . . . . 5 (𝑟 = 𝑆 𝑟 = 𝑆)
21eqeq2d 2748 . . . 4 (𝑟 = 𝑆 → (𝑋 = 𝑟𝑋 = 𝑆))
3 pweq 4614 . . . . . 6 (𝑟 = 𝑆 → 𝒫 𝑟 = 𝒫 𝑆)
43ineq1d 4219 . . . . 5 (𝑟 = 𝑆 → (𝒫 𝑟 ∩ Fin) = (𝒫 𝑆 ∩ Fin))
54rexeqdv 3327 . . . 4 (𝑟 = 𝑆 → (∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = 𝑠 ↔ ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = 𝑠))
62, 5imbi12d 344 . . 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
87iscmp 23396 . . . . 5 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑟 ∈ 𝒫 𝐽(𝑋 = 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = 𝑠)))
98simprbi 496 . . . 4 (𝐽 ∈ Comp → ∀𝑟 ∈ 𝒫 𝐽(𝑋 = 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = 𝑠))
109adantr 480 . . 3 ((𝐽 ∈ Comp ∧ 𝑆𝐽) → ∀𝑟 ∈ 𝒫 𝐽(𝑋 = 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = 𝑠))
11 ssexg 5323 . . . . 5 ((𝑆𝐽𝐽 ∈ Comp) → 𝑆 ∈ V)
1211ancoms 458 . . . 4 ((𝐽 ∈ Comp ∧ 𝑆𝐽) → 𝑆 ∈ V)
13 simpr 484 . . . 4 ((𝐽 ∈ Comp ∧ 𝑆𝐽) → 𝑆𝐽)
1412, 13elpwd 4606 . . 3 ((𝐽 ∈ Comp ∧ 𝑆𝐽) → 𝑆 ∈ 𝒫 𝐽)
156, 10, 14rspcdva 3623 . 2 ((𝐽 ∈ Comp ∧ 𝑆𝐽) → (𝑋 = 𝑆 → ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = 𝑠))
16153impia 1118 1 ((𝐽 ∈ Comp ∧ 𝑆𝐽𝑋 = 𝑆) → ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = 𝑠)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  cin 3950  wss 3951  𝒫 cpw 4600   cuni 4907  Fincfn 8985  Topctop 22899  Compccmp 23394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-in 3958  df-ss 3968  df-pw 4602  df-uni 4908  df-cmp 23395
This theorem is referenced by:  cmpcov2  23398  cncmp  23400  discmp  23406  cmpcld  23410  sscmp  23413  comppfsc  23540  alexsubALTlem1  24055  ptcmplem3  24062  lebnum  24996  heibor1  37817
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