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Theorem fincmp 21929
Description: A finite topology is compact. (Contributed by FL, 22-Dec-2008.)
Assertion
Ref Expression
fincmp (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Comp)

Proof of Theorem fincmp
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elinel1 4169 . 2 (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Top)
2 elinel2 4170 . . 3 (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Fin)
3 vex 3495 . . . . . 6 𝑦 ∈ V
43pwid 4554 . . . . 5 𝑦 ∈ 𝒫 𝑦
5 velpw 4543 . . . . . 6 (𝑦 ∈ 𝒫 𝐽𝑦𝐽)
6 ssfi 8726 . . . . . 6 ((𝐽 ∈ Fin ∧ 𝑦𝐽) → 𝑦 ∈ Fin)
75, 6sylan2b 593 . . . . 5 ((𝐽 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝐽) → 𝑦 ∈ Fin)
8 elin 4166 . . . . . 6 (𝑦 ∈ (𝒫 𝑦 ∩ Fin) ↔ (𝑦 ∈ 𝒫 𝑦𝑦 ∈ Fin))
9 unieq 4838 . . . . . . . 8 (𝑧 = 𝑦 𝑧 = 𝑦)
109rspceeqv 3635 . . . . . . 7 ((𝑦 ∈ (𝒫 𝑦 ∩ Fin) ∧ 𝐽 = 𝑦) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧)
1110ex 413 . . . . . 6 (𝑦 ∈ (𝒫 𝑦 ∩ Fin) → ( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧))
128, 11sylbir 236 . . . . 5 ((𝑦 ∈ 𝒫 𝑦𝑦 ∈ Fin) → ( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧))
134, 7, 12sylancr 587 . . . 4 ((𝐽 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝐽) → ( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧))
1413ralrimiva 3179 . . 3 (𝐽 ∈ Fin → ∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧))
152, 14syl 17 . 2 (𝐽 ∈ (Top ∩ Fin) → ∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧))
16 eqid 2818 . . 3 𝐽 = 𝐽
1716iscmp 21924 . 2 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧)))
181, 15, 17sylanbrc 583 1 (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136  cin 3932  wss 3933  𝒫 cpw 4535   cuni 4830  Fincfn 8497  Topctop 21429  Compccmp 21922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-om 7570  df-er 8278  df-en 8498  df-fin 8501  df-cmp 21923
This theorem is referenced by:  0cmp  21930  discmp  21934  1stckgenlem  22089  ptcmpfi  22349  kelac2lem  39542
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