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Theorem fincmp 21244
 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 inss1 3866 . . 3 (Top ∩ Fin) ⊆ Top
21sseli 3632 . 2 (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Top)
3 inss2 3867 . . . 4 (Top ∩ Fin) ⊆ Fin
43sseli 3632 . . 3 (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Fin)
5 vex 3234 . . . . . 6 𝑦 ∈ V
65pwid 4207 . . . . 5 𝑦 ∈ 𝒫 𝑦
7 selpw 4198 . . . . . 6 (𝑦 ∈ 𝒫 𝐽𝑦𝐽)
8 ssfi 8221 . . . . . 6 ((𝐽 ∈ Fin ∧ 𝑦𝐽) → 𝑦 ∈ Fin)
97, 8sylan2b 491 . . . . 5 ((𝐽 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝐽) → 𝑦 ∈ Fin)
10 elin 3829 . . . . . 6 (𝑦 ∈ (𝒫 𝑦 ∩ Fin) ↔ (𝑦 ∈ 𝒫 𝑦𝑦 ∈ Fin))
11 unieq 4476 . . . . . . . . 9 (𝑧 = 𝑦 𝑧 = 𝑦)
1211eqeq2d 2661 . . . . . . . 8 (𝑧 = 𝑦 → ( 𝐽 = 𝑧 𝐽 = 𝑦))
1312rspcev 3340 . . . . . . 7 ((𝑦 ∈ (𝒫 𝑦 ∩ Fin) ∧ 𝐽 = 𝑦) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧)
1413ex 449 . . . . . 6 (𝑦 ∈ (𝒫 𝑦 ∩ Fin) → ( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧))
1510, 14sylbir 225 . . . . 5 ((𝑦 ∈ 𝒫 𝑦𝑦 ∈ Fin) → ( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧))
166, 9, 15sylancr 696 . . . 4 ((𝐽 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝐽) → ( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧))
1716ralrimiva 2995 . . 3 (𝐽 ∈ Fin → ∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧))
184, 17syl 17 . 2 (𝐽 ∈ (Top ∩ Fin) → ∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧))
19 eqid 2651 . . 3 𝐽 = 𝐽
2019iscmp 21239 . 2 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧)))
212, 18, 20sylanbrc 699 1 (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Comp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942   ∩ cin 3606   ⊆ wss 3607  𝒫 cpw 4191  ∪ cuni 4468  Fincfn 7997  Topctop 20746  Compccmp 21237 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-om 7108  df-er 7787  df-en 7998  df-fin 8001  df-cmp 21238 This theorem is referenced by:  0cmp  21245  discmp  21249  1stckgenlem  21404  ptcmpfi  21664  kelac2lem  37951
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