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Theorem iscmp 21412
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 4300 . . 3 (𝑥 = 𝐽 → 𝒫 𝑥 = 𝒫 𝐽)
2 unieq 4582 . . . . . 6 (𝑥 = 𝐽 𝑥 = 𝐽)
3 iscmp.1 . . . . . 6 𝑋 = 𝐽
42, 3syl6eqr 2823 . . . . 5 (𝑥 = 𝐽 𝑥 = 𝑋)
54eqeq1d 2773 . . . 4 (𝑥 = 𝐽 → ( 𝑥 = 𝑦𝑋 = 𝑦))
64eqeq1d 2773 . . . . 5 (𝑥 = 𝐽 → ( 𝑥 = 𝑧𝑋 = 𝑧))
76rexbidv 3200 . . . 4 (𝑥 = 𝐽 → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝑥 = 𝑧 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧))
85, 7imbi12d 333 . . 3 (𝑥 = 𝐽 → (( 𝑥 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝑥 = 𝑧) ↔ (𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)))
91, 8raleqbidv 3301 . 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 21411 . 2 Comp = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥( 𝑥 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝑥 = 𝑧)}
119, 10elrab2 3518 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 382   = wceq 1631  wcel 2145  wral 3061  wrex 3062  cin 3722  𝒫 cpw 4297   cuni 4574  Fincfn 8109  Topctop 20918  Compccmp 21410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-in 3730  df-ss 3737  df-pw 4299  df-uni 4575  df-cmp 21411
This theorem is referenced by:  cmpcov  21413  cncmp  21416  fincmp  21417  cmptop  21419  cmpsub  21424  tgcmp  21425  uncmp  21427  sscmp  21429  cmpfi  21432  comppfsc  21556  txcmp  21667  alexsubb  22070  alexsubALT  22075  cmpcref  30257  onsucsuccmpi  32779  limsucncmpi  32781  heibor  33952
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