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Theorem fincmp 21101
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 3816 . . 3 (Top ∩ Fin) ⊆ Top
21sseli 3584 . 2 (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Top)
3 inss2 3817 . . . 4 (Top ∩ Fin) ⊆ Fin
43sseli 3584 . . 3 (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Fin)
5 vex 3194 . . . . . 6 𝑦 ∈ V
65pwid 4150 . . . . 5 𝑦 ∈ 𝒫 𝑦
7 selpw 4142 . . . . . 6 (𝑦 ∈ 𝒫 𝐽𝑦𝐽)
8 ssfi 8125 . . . . . 6 ((𝐽 ∈ Fin ∧ 𝑦𝐽) → 𝑦 ∈ Fin)
97, 8sylan2b 492 . . . . 5 ((𝐽 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝐽) → 𝑦 ∈ Fin)
10 elin 3779 . . . . . 6 (𝑦 ∈ (𝒫 𝑦 ∩ Fin) ↔ (𝑦 ∈ 𝒫 𝑦𝑦 ∈ Fin))
11 unieq 4415 . . . . . . . . 9 (𝑧 = 𝑦 𝑧 = 𝑦)
1211eqeq2d 2636 . . . . . . . 8 (𝑧 = 𝑦 → ( 𝐽 = 𝑧 𝐽 = 𝑦))
1312rspcev 3300 . . . . . . 7 ((𝑦 ∈ (𝒫 𝑦 ∩ Fin) ∧ 𝐽 = 𝑦) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧)
1413ex 450 . . . . . 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 694 . . . 4 ((𝐽 ∈ Fin ∧ 𝑦 ∈ 𝒫 𝐽) → ( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧))
1716ralrimiva 2965 . . 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 2626 . . 3 𝐽 = 𝐽
2019iscmp 21096 . 2 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧)))
212, 18, 20sylanbrc 697 1 (𝐽 ∈ (Top ∩ Fin) → 𝐽 ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  wral 2912  wrex 2913  cin 3559  wss 3560  𝒫 cpw 4135   cuni 4407  Fincfn 7900  Topctop 20612  Compccmp 21094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-om 7014  df-er 7688  df-en 7901  df-fin 7904  df-cmp 21095
This theorem is referenced by:  0cmp  21102  discmp  21106  1stckgenlem  21261  ptcmpfi  21521  kelac2lem  37100
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