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Theorem iscmp 22737
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 4574 . . 3 (𝑥 = 𝐽 → 𝒫 𝑥 = 𝒫 𝐽)
2 unieq 4876 . . . . . 6 (𝑥 = 𝐽 𝑥 = 𝐽)
3 iscmp.1 . . . . . 6 𝑋 = 𝐽
42, 3eqtr4di 2794 . . . . 5 (𝑥 = 𝐽 𝑥 = 𝑋)
54eqeq1d 2738 . . . 4 (𝑥 = 𝐽 → ( 𝑥 = 𝑦𝑋 = 𝑦))
64eqeq1d 2738 . . . . 5 (𝑥 = 𝐽 → ( 𝑥 = 𝑧𝑋 = 𝑧))
76rexbidv 3175 . . . 4 (𝑥 = 𝐽 → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝑥 = 𝑧 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧))
85, 7imbi12d 344 . . 3 (𝑥 = 𝐽 → (( 𝑥 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝑥 = 𝑧) ↔ (𝑋 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = 𝑧)))
91, 8raleqbidv 3319 . 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 22736 . 2 Comp = {𝑥 ∈ Top ∣ ∀𝑦 ∈ 𝒫 𝑥( 𝑥 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝑥 = 𝑧)}
119, 10elrab2 3648 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 205  wa 396   = wceq 1541  wcel 2106  wral 3064  wrex 3073  cin 3909  𝒫 cpw 4560   cuni 4865  Fincfn 8882  Topctop 22240  Compccmp 22735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-in 3917  df-ss 3927  df-pw 4562  df-uni 4866  df-cmp 22736
This theorem is referenced by:  cmpcov  22738  cncmp  22741  fincmp  22742  cmptop  22744  cmpsub  22749  tgcmp  22750  uncmp  22752  sscmp  22754  cmpfi  22757  comppfsc  22881  txcmp  22992  alexsubb  23395  alexsubALT  23400  cmpcref  32371  onsucsuccmpi  34905  limsucncmpi  34907  pibp16  35874  heibor  36270
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