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Theorem sscmp 21536
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 21045 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
213ad2ant1 1164 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) → 𝐽 ∈ Top)
3 elpwi 4360 . . . 4 (𝑥 ∈ 𝒫 𝐽𝑥𝐽)
4 simpl2 1245 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝐾 ∈ Comp)
5 simprl 788 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝑥𝐽)
6 simpl3 1247 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝐽𝐾)
75, 6sstrd 3809 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝑥𝐾)
8 simpl1 1243 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝐽 ∈ (TopOn‘𝑋))
9 toponuni 21046 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
108, 9syl 17 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝑋 = 𝐽)
11 simprr 790 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝐽 = 𝑥)
1210, 11eqtrd 2834 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝑋 = 𝑥)
13 sscmp.1 . . . . . . . 8 𝑋 = 𝐾
1413cmpcov 21520 . . . . . . 7 ((𝐾 ∈ Comp ∧ 𝑥𝐾𝑋 = 𝑥) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)
154, 7, 12, 14syl3anc 1491 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)
1610eqeq1d 2802 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → (𝑋 = 𝑦 𝐽 = 𝑦))
1716rexbidv 3234 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → (∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦 ↔ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin) 𝐽 = 𝑦))
1815, 17mpbid 224 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin) 𝐽 = 𝑦)
1918expr 449 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ 𝑥𝐽) → ( 𝐽 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin) 𝐽 = 𝑦))
203, 19sylan2 587 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ 𝑥 ∈ 𝒫 𝐽) → ( 𝐽 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin) 𝐽 = 𝑦))
2120ralrimiva 3148 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) → ∀𝑥 ∈ 𝒫 𝐽( 𝐽 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin) 𝐽 = 𝑦))
22 eqid 2800 . . 3 𝐽 = 𝐽
2322iscmp 21519 . 2 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝐽( 𝐽 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin) 𝐽 = 𝑦)))
242, 21, 23sylanbrc 579 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) → 𝐽 ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108   = wceq 1653  wcel 2157  wral 3090  wrex 3091  cin 3769  wss 3770  𝒫 cpw 4350   cuni 4629  cfv 6102  Fincfn 8196  Topctop 21025  TopOnctopon 21042  Compccmp 21517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-iota 6065  df-fun 6104  df-fv 6110  df-topon 21043  df-cmp 21518
This theorem is referenced by:  kgencmp2  21677  kgen2ss  21686
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