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Theorem cmpcov 23381
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 4916 . . . . 5 (𝑟 = 𝑆 𝑟 = 𝑆)
21eqeq2d 2737 . . . 4 (𝑟 = 𝑆 → (𝑋 = 𝑟𝑋 = 𝑆))
3 pweq 4611 . . . . . 6 (𝑟 = 𝑆 → 𝒫 𝑟 = 𝒫 𝑆)
43ineq1d 4209 . . . . 5 (𝑟 = 𝑆 → (𝒫 𝑟 ∩ Fin) = (𝒫 𝑆 ∩ Fin))
54rexeqdv 3316 . . . 4 (𝑟 = 𝑆 → (∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = 𝑠 ↔ ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = 𝑠))
62, 5imbi12d 343 . . 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 23380 . . . . 5 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑟 ∈ 𝒫 𝐽(𝑋 = 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = 𝑠)))
98simprbi 495 . . . 4 (𝐽 ∈ Comp → ∀𝑟 ∈ 𝒫 𝐽(𝑋 = 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = 𝑠))
109adantr 479 . . 3 ((𝐽 ∈ Comp ∧ 𝑆𝐽) → ∀𝑟 ∈ 𝒫 𝐽(𝑋 = 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = 𝑠))
11 ssexg 5320 . . . . 5 ((𝑆𝐽𝐽 ∈ Comp) → 𝑆 ∈ V)
1211ancoms 457 . . . 4 ((𝐽 ∈ Comp ∧ 𝑆𝐽) → 𝑆 ∈ V)
13 simpr 483 . . . 4 ((𝐽 ∈ Comp ∧ 𝑆𝐽) → 𝑆𝐽)
1412, 13elpwd 4603 . . 3 ((𝐽 ∈ Comp ∧ 𝑆𝐽) → 𝑆 ∈ 𝒫 𝐽)
156, 10, 14rspcdva 3608 . 2 ((𝐽 ∈ Comp ∧ 𝑆𝐽) → (𝑋 = 𝑆 → ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = 𝑠))
16153impia 1114 1 ((𝐽 ∈ Comp ∧ 𝑆𝐽𝑋 = 𝑆) → ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = 𝑠)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1534  wcel 2099  wral 3051  wrex 3060  Vcvv 3462  cin 3945  wss 3946  𝒫 cpw 4597   cuni 4905  Fincfn 8966  Topctop 22883  Compccmp 23378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5296
This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1086  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-in 3953  df-ss 3963  df-pw 4599  df-uni 4906  df-cmp 23379
This theorem is referenced by:  cmpcov2  23382  cncmp  23384  discmp  23390  cmpcld  23394  sscmp  23397  comppfsc  23524  alexsubALTlem1  24039  ptcmplem3  24046  lebnum  24978  heibor1  37524
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