MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cmpcov Structured version   Visualization version   GIF version

Theorem cmpcov 21097
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 4415 . . . . 5 (𝑟 = 𝑆 𝑟 = 𝑆)
21eqeq2d 2636 . . . 4 (𝑟 = 𝑆 → (𝑋 = 𝑟𝑋 = 𝑆))
3 pweq 4138 . . . . . 6 (𝑟 = 𝑆 → 𝒫 𝑟 = 𝒫 𝑆)
43ineq1d 3796 . . . . 5 (𝑟 = 𝑆 → (𝒫 𝑟 ∩ Fin) = (𝒫 𝑆 ∩ Fin))
54rexeqdv 3139 . . . 4 (𝑟 = 𝑆 → (∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = 𝑠 ↔ ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = 𝑠))
62, 5imbi12d 334 . . 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 21096 . . . . 5 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑟 ∈ 𝒫 𝐽(𝑋 = 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = 𝑠)))
98simprbi 480 . . . 4 (𝐽 ∈ Comp → ∀𝑟 ∈ 𝒫 𝐽(𝑋 = 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = 𝑠))
109adantr 481 . . 3 ((𝐽 ∈ Comp ∧ 𝑆𝐽) → ∀𝑟 ∈ 𝒫 𝐽(𝑋 = 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑋 = 𝑠))
11 simpr 477 . . . 4 ((𝐽 ∈ Comp ∧ 𝑆𝐽) → 𝑆𝐽)
12 ssexg 4769 . . . . . 6 ((𝑆𝐽𝐽 ∈ Comp) → 𝑆 ∈ V)
1312ancoms 469 . . . . 5 ((𝐽 ∈ Comp ∧ 𝑆𝐽) → 𝑆 ∈ V)
14 elpwg 4143 . . . . 5 (𝑆 ∈ V → (𝑆 ∈ 𝒫 𝐽𝑆𝐽))
1513, 14syl 17 . . . 4 ((𝐽 ∈ Comp ∧ 𝑆𝐽) → (𝑆 ∈ 𝒫 𝐽𝑆𝐽))
1611, 15mpbird 247 . . 3 ((𝐽 ∈ Comp ∧ 𝑆𝐽) → 𝑆 ∈ 𝒫 𝐽)
176, 10, 16rspcdva 3306 . 2 ((𝐽 ∈ Comp ∧ 𝑆𝐽) → (𝑋 = 𝑆 → ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = 𝑠))
18173impia 1258 1 ((𝐽 ∈ Comp ∧ 𝑆𝐽𝑋 = 𝑆) → ∃𝑠 ∈ (𝒫 𝑆 ∩ Fin)𝑋 = 𝑠)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  wral 2912  wrex 2913  Vcvv 3191  cin 3559  wss 3560  𝒫 cpw 4135   cuni 4407  Fincfn 7900  Topctop 20612  Compccmp 21094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-in 3567  df-ss 3574  df-pw 4137  df-uni 4408  df-cmp 21095
This theorem is referenced by:  cmpcov2  21098  cncmp  21100  discmp  21106  cmpcld  21110  sscmp  21113  comppfsc  21240  alexsubALTlem1  21756  ptcmplem3  21763  lebnum  22666  heibor1  33227
  Copyright terms: Public domain W3C validator