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Theorem sscmp 22008
Description: A subset of a compact topology (i.e. a coarser topology) is compact. (Contributed by Mario Carneiro, 20-Mar-2015.)
Hypothesis
Ref Expression
sscmp.1 𝑋 = 𝐾
Assertion
Ref Expression
sscmp ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) → 𝐽 ∈ Comp)

Proof of Theorem sscmp
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 topontop 21516 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
213ad2ant1 1130 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) → 𝐽 ∈ Top)
3 elpwi 4520 . . . 4 (𝑥 ∈ 𝒫 𝐽𝑥𝐽)
4 simpl2 1189 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝐾 ∈ Comp)
5 simprl 770 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝑥𝐽)
6 simpl3 1190 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝐽𝐾)
75, 6sstrd 3952 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝑥𝐾)
8 simpl1 1188 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝐽 ∈ (TopOn‘𝑋))
9 toponuni 21517 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
108, 9syl 17 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝑋 = 𝐽)
11 simprr 772 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝐽 = 𝑥)
1210, 11eqtrd 2857 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝑋 = 𝑥)
13 sscmp.1 . . . . . . . 8 𝑋 = 𝐾
1413cmpcov 21992 . . . . . . 7 ((𝐾 ∈ Comp ∧ 𝑥𝐾𝑋 = 𝑥) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)
154, 7, 12, 14syl3anc 1368 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)
1610eqeq1d 2824 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → (𝑋 = 𝑦 𝐽 = 𝑦))
1716rexbidv 3283 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → (∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦 ↔ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin) 𝐽 = 𝑦))
1815, 17mpbid 235 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin) 𝐽 = 𝑦)
1918expr 460 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ 𝑥𝐽) → ( 𝐽 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin) 𝐽 = 𝑦))
203, 19sylan2 595 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ 𝑥 ∈ 𝒫 𝐽) → ( 𝐽 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin) 𝐽 = 𝑦))
2120ralrimiva 3174 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) → ∀𝑥 ∈ 𝒫 𝐽( 𝐽 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin) 𝐽 = 𝑦))
22 eqid 2822 . . 3 𝐽 = 𝐽
2322iscmp 21991 . 2 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝐽( 𝐽 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin) 𝐽 = 𝑦)))
242, 21, 23sylanbrc 586 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) → 𝐽 ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2114  wral 3130  wrex 3131  cin 3907  wss 3908  𝒫 cpw 4511   cuni 4813  cfv 6334  Fincfn 8496  Topctop 21496  TopOnctopon 21513  Compccmp 21989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-topon 21514  df-cmp 21990
This theorem is referenced by:  kgencmp2  22149  kgen2ss  22158
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