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Theorem cmpcov 21002
 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 simpr 476 . . . 4 ((𝐽 ∈ Comp ∧ 𝑆𝐽) → 𝑆𝐽)
2 ssexg 4732 . . . . . 6 ((𝑆𝐽𝐽 ∈ Comp) → 𝑆 ∈ V)
32ancoms 468 . . . . 5 ((𝐽 ∈ Comp ∧ 𝑆𝐽) → 𝑆 ∈ V)
4 elpwg 4116 . . . . 5 (𝑆 ∈ V → (𝑆 ∈ 𝒫 𝐽𝑆𝐽))
53, 4syl 17 . . . 4 ((𝐽 ∈ Comp ∧ 𝑆𝐽) → (𝑆 ∈ 𝒫 𝐽𝑆𝐽))
61, 5mpbird 246 . . 3 ((𝐽 ∈ Comp ∧ 𝑆𝐽) → 𝑆 ∈ 𝒫 𝐽)
7 iscmp.1 . . . . . 6 𝑋 = 𝐽
87iscmp 21001 . . . . 5 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑟 ∈ 𝒫 𝐽(𝑋 = 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = 𝑠)))
98simprbi 479 . . . 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 unieq 4380 . . . . . 6 (𝑟 = 𝑆 𝑟 = 𝑆)
1211eqeq2d 2620 . . . . 5 (𝑟 = 𝑆 → (𝑋 = 𝑟𝑋 = 𝑆))
13 pweq 4111 . . . . . . 7 (𝑟 = 𝑆 → 𝒫 𝑟 = 𝒫 𝑆)
1413ineq1d 3775 . . . . . 6 (𝑟 = 𝑆 → (𝒫 𝑟 ∩ Fin) = (𝒫 𝑆 ∩ Fin))
1514rexeqdv 3122 . . . . 5 (𝑟 = 𝑆 → (∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = 𝑠 ↔ ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = 𝑠))
1612, 15imbi12d 333 . . . 4 (𝑟 = 𝑆 → ((𝑋 = 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = 𝑠) ↔ (𝑋 = 𝑆 → ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = 𝑠)))
1716rspcv 3278 . . 3 (𝑆 ∈ 𝒫 𝐽 → (∀𝑟 ∈ 𝒫 𝐽(𝑋 = 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = 𝑠) → (𝑋 = 𝑆 → ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = 𝑠)))
186, 10, 17sylc 63 . 2 ((𝐽 ∈ Comp ∧ 𝑆𝐽) → (𝑋 = 𝑆 → ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = 𝑠))
19183impia 1253 1 ((𝐽 ∈ Comp ∧ 𝑆𝐽𝑋 = 𝑆) → ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = 𝑠)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  Fincfn 7841  Topctop 20517  Compccmp 20999 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110  df-uni 4373  df-cmp 21000 This theorem is referenced by:  cmpcov2  21003  cncmp  21005  discmp  21011  cmpcld  21015  sscmp  21018  comppfsc  21145  alexsubALTlem1  21661  ptcmplem3  21668  lebnum  22571  heibor1  32779
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