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Theorem iscmp 21104
Description: The predicate "is a compact topology". (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 11-Feb-2015.)
Hypothesis
Ref Expression
iscmp.1 𝑋 = 𝐽
Assertion
Ref Expression
iscmp (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)))
Distinct variable group:   𝑦,𝑧,𝐽
Allowed substitution hints:   𝑋(𝑦,𝑧)

Proof of Theorem iscmp
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pweq 4135 . . 3 (𝑥 = 𝐽 → 𝒫 𝑥 = 𝒫 𝐽)
2 unieq 4412 . . . . . 6 (𝑥 = 𝐽 𝑥 = 𝐽)
3 iscmp.1 . . . . . 6 𝑋 = 𝐽
42, 3syl6eqr 2673 . . . . 5 (𝑥 = 𝐽 𝑥 = 𝑋)
54eqeq1d 2623 . . . 4 (𝑥 = 𝐽 → ( 𝑥 = 𝑦𝑋 = 𝑦))
64eqeq1d 2623 . . . . 5 (𝑥 = 𝐽 → ( 𝑥 = 𝑧𝑋 = 𝑧))
76rexbidv 3045 . . . 4 (𝑥 = 𝐽 → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝑥 = 𝑧 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧))
85, 7imbi12d 334 . . 3 (𝑥 = 𝐽 → (( 𝑥 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝑥 = 𝑧) ↔ (𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)))
91, 8raleqbidv 3141 . 2 (𝑥 = 𝐽 → (∀𝑦 ∈ 𝒫 𝑥( 𝑥 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝑥 = 𝑧) ↔ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)))
10 df-cmp 21103 . 2 Comp = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥( 𝑥 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝑥 = 𝑧)}
119, 10elrab2 3349 1 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  cin 3555  𝒫 cpw 4132   cuni 4404  Fincfn 7902  Topctop 20620  Compccmp 21102
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 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-in 3563  df-ss 3570  df-pw 4134  df-uni 4405  df-cmp 21103
This theorem is referenced by:  cmpcov  21105  cncmp  21108  fincmp  21109  cmptop  21111  cmpsub  21116  tgcmp  21117  uncmp  21119  sscmp  21121  cmpfi  21124  comppfsc  21248  txcmp  21359  alexsubb  21763  alexsubALT  21768  cmpcref  29711  onsucsuccmpi  32105  limsucncmpi  32107  heibor  33273
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