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Theorem fincmp 22542
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 4134 . 2 (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Top)
2 elinel2 4135 . . 3 (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Fin)
3 vex 3435 . . . . . 6 𝑦 ∈ V
43pwid 4563 . . . . 5 𝑦 ∈ 𝒫 𝑦
5 velpw 4544 . . . . . 6 (𝑦 ∈ 𝒫 𝐽𝑦𝐽)
6 ssfi 8938 . . . . . 6 ((𝐽 ∈ Fin ∧ 𝑦𝐽) → 𝑦 ∈ Fin)
75, 6sylan2b 594 . . . . 5 ((𝐽 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝐽) → 𝑦 ∈ Fin)
8 elin 3908 . . . . . 6 (𝑦 ∈ (𝒫 𝑦 ∩ Fin) ↔ (𝑦 ∈ 𝒫 𝑦𝑦 ∈ Fin))
9 unieq 4856 . . . . . . . 8 (𝑧 = 𝑦 𝑧 = 𝑦)
109rspceeqv 3576 . . . . . . 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 234 . . . . 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 3110 . . 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 2740 . . 3 𝐽 = 𝐽
1716iscmp 22537 . 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 1542  wcel 2110  wral 3066  wrex 3067  cin 3891  wss 3892  𝒫 cpw 4539   cuni 4845  Fincfn 8716  Topctop 22040  Compccmp 22535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356  ax-un 7582
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-om 7707  df-1o 8288  df-en 8717  df-fin 8720  df-cmp 22536
This theorem is referenced by:  0cmp  22543  discmp  22547  1stckgenlem  22702  ptcmpfi  22962  kelac2lem  40886
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