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Theorem sscmp 23429
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 22935 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
213ad2ant1 1132 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) → 𝐽 ∈ Top)
3 elpwi 4612 . . . 4 (𝑥 ∈ 𝒫 𝐽𝑥𝐽)
4 simpl2 1191 . . . . . . 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 1192 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝐽𝐾)
75, 6sstrd 4006 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝑥𝐾)
8 simpl1 1190 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝐽 ∈ (TopOn‘𝑋))
9 toponuni 22936 . . . . . . . . 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 2775 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝑋 = 𝑥)
13 sscmp.1 . . . . . . . 8 𝑋 = 𝐾
1413cmpcov 23413 . . . . . . 7 ((𝐾 ∈ Comp ∧ 𝑥𝐾𝑋 = 𝑥) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)
154, 7, 12, 14syl3anc 1370 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)
1610eqeq1d 2737 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → (𝑋 = 𝑦 𝐽 = 𝑦))
1716rexbidv 3177 . . . . . 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 3144 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) → ∀𝑥 ∈ 𝒫 𝐽( 𝐽 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin) 𝐽 = 𝑦))
22 eqid 2735 . . 3 𝐽 = 𝐽
2322iscmp 23412 . 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 1086   = wceq 1537  wcel 2106  wral 3059  wrex 3068  cin 3962  wss 3963  𝒫 cpw 4605   cuni 4912  cfv 6563  Fincfn 8984  Topctop 22915  TopOnctopon 22932  Compccmp 23410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-topon 22933  df-cmp 23411
This theorem is referenced by:  kgencmp2  23570  kgen2ss  23579
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