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Theorem sscmp 23413
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 22919 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
213ad2ant1 1134 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) → 𝐽 ∈ Top)
3 elpwi 4607 . . . 4 (𝑥 ∈ 𝒫 𝐽𝑥𝐽)
4 simpl2 1193 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝐾 ∈ Comp)
5 simprl 771 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝑥𝐽)
6 simpl3 1194 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝐽𝐾)
75, 6sstrd 3994 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝑥𝐾)
8 simpl1 1192 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝐽 ∈ (TopOn‘𝑋))
9 toponuni 22920 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
108, 9syl 17 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝑋 = 𝐽)
11 simprr 773 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝐽 = 𝑥)
1210, 11eqtrd 2777 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝑋 = 𝑥)
13 sscmp.1 . . . . . . . 8 𝑋 = 𝐾
1413cmpcov 23397 . . . . . . 7 ((𝐾 ∈ Comp ∧ 𝑥𝐾𝑋 = 𝑥) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)
154, 7, 12, 14syl3anc 1373 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)
1610eqeq1d 2739 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → (𝑋 = 𝑦 𝐽 = 𝑦))
1716rexbidv 3179 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → (∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦 ↔ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin) 𝐽 = 𝑦))
1815, 17mpbid 232 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin) 𝐽 = 𝑦)
1918expr 456 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ 𝑥𝐽) → ( 𝐽 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin) 𝐽 = 𝑦))
203, 19sylan2 593 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ 𝑥 ∈ 𝒫 𝐽) → ( 𝐽 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin) 𝐽 = 𝑦))
2120ralrimiva 3146 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) → ∀𝑥 ∈ 𝒫 𝐽( 𝐽 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin) 𝐽 = 𝑦))
22 eqid 2737 . . 3 𝐽 = 𝐽
2322iscmp 23396 . 2 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝐽( 𝐽 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin) 𝐽 = 𝑦)))
242, 21, 23sylanbrc 583 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) → 𝐽 ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  cin 3950  wss 3951  𝒫 cpw 4600   cuni 4907  cfv 6561  Fincfn 8985  Topctop 22899  TopOnctopon 22916  Compccmp 23394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-topon 22917  df-cmp 23395
This theorem is referenced by:  kgencmp2  23554  kgen2ss  23563
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