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Theorem fincmp 22767
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 4159 . 2 (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Top)
2 elinel2 4160 . . 3 (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Fin)
3 vex 3451 . . . . . 6 𝑦 ∈ V
43pwid 4586 . . . . 5 𝑦 ∈ 𝒫 𝑦
5 velpw 4569 . . . . . 6 (𝑦 ∈ 𝒫 𝐽𝑦𝐽)
6 ssfi 9123 . . . . . 6 ((𝐽 ∈ Fin ∧ 𝑦𝐽) → 𝑦 ∈ Fin)
75, 6sylan2b 595 . . . . 5 ((𝐽 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝐽) → 𝑦 ∈ Fin)
8 elin 3930 . . . . . 6 (𝑦 ∈ (𝒫 𝑦 ∩ Fin) ↔ (𝑦 ∈ 𝒫 𝑦𝑦 ∈ Fin))
9 unieq 4880 . . . . . . . 8 (𝑧 = 𝑦 𝑧 = 𝑦)
109rspceeqv 3599 . . . . . . 7 ((𝑦 ∈ (𝒫 𝑦 ∩ Fin) ∧ 𝐽 = 𝑦) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧)
1110ex 414 . . . . . 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 588 . . . 4 ((𝐽 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝐽) → ( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧))
1413ralrimiva 3140 . . 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 2733 . . 3 𝐽 = 𝐽
1716iscmp 22762 . 2 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧)))
181, 15, 17sylanbrc 584 1 (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3061  wrex 3070  cin 3913  wss 3914  𝒫 cpw 4564   cuni 4869  Fincfn 8889  Topctop 22265  Compccmp 22760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-om 7807  df-1o 8416  df-en 8890  df-fin 8893  df-cmp 22761
This theorem is referenced by:  0cmp  22768  discmp  22772  1stckgenlem  22927  ptcmpfi  23187  kelac2lem  41438
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