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Theorem cmpcov 22756
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 4877 . . . . 5 (𝑟 = 𝑆 𝑟 = 𝑆)
21eqeq2d 2744 . . . 4 (𝑟 = 𝑆 → (𝑋 = 𝑟𝑋 = 𝑆))
3 pweq 4575 . . . . . 6 (𝑟 = 𝑆 → 𝒫 𝑟 = 𝒫 𝑆)
43ineq1d 4172 . . . . 5 (𝑟 = 𝑆 → (𝒫 𝑟 ∩ Fin) = (𝒫 𝑆 ∩ Fin))
54rexeqdv 3313 . . . 4 (𝑟 = 𝑆 → (∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = 𝑠 ↔ ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = 𝑠))
62, 5imbi12d 345 . . 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 22755 . . . . 5 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑟 ∈ 𝒫 𝐽(𝑋 = 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = 𝑠)))
98simprbi 498 . . . 4 (𝐽 ∈ Comp → ∀𝑟 ∈ 𝒫 𝐽(𝑋 = 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = 𝑠))
109adantr 482 . . 3 ((𝐽 ∈ Comp ∧ 𝑆𝐽) → ∀𝑟 ∈ 𝒫 𝐽(𝑋 = 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = 𝑠))
11 ssexg 5281 . . . . 5 ((𝑆𝐽𝐽 ∈ Comp) → 𝑆 ∈ V)
1211ancoms 460 . . . 4 ((𝐽 ∈ Comp ∧ 𝑆𝐽) → 𝑆 ∈ V)
13 simpr 486 . . . 4 ((𝐽 ∈ Comp ∧ 𝑆𝐽) → 𝑆𝐽)
1412, 13elpwd 4567 . . 3 ((𝐽 ∈ Comp ∧ 𝑆𝐽) → 𝑆 ∈ 𝒫 𝐽)
156, 10, 14rspcdva 3581 . 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 397  w3a 1088   = wceq 1542  wcel 2107  wral 3061  wrex 3070  Vcvv 3444  cin 3910  wss 3911  𝒫 cpw 4561   cuni 4866  Fincfn 8886  Topctop 22258  Compccmp 22753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257
This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-in 3918  df-ss 3928  df-pw 4563  df-uni 4867  df-cmp 22754
This theorem is referenced by:  cmpcov2  22757  cncmp  22759  discmp  22765  cmpcld  22769  sscmp  22772  comppfsc  22899  alexsubALTlem1  23414  ptcmplem3  23421  lebnum  24343  heibor1  36315
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