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Theorem sscmp 21131
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 20650 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
213ad2ant1 1080 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) → 𝐽 ∈ Top)
3 elpwi 4145 . . . 4 (𝑥 ∈ 𝒫 𝐽𝑥𝐽)
4 simpl2 1063 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝐾 ∈ Comp)
5 simprl 793 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝑥𝐽)
6 simpl3 1064 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝐽𝐾)
75, 6sstrd 3597 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝑥𝐾)
8 simpl1 1062 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝐽 ∈ (TopOn‘𝑋))
9 toponuni 20651 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
108, 9syl 17 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝑋 = 𝐽)
11 simprr 795 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝐽 = 𝑥)
1210, 11eqtrd 2655 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → 𝑋 = 𝑥)
13 sscmp.1 . . . . . . . 8 𝑋 = 𝐾
1413cmpcov 21115 . . . . . . 7 ((𝐾 ∈ Comp ∧ 𝑥𝐾𝑋 = 𝑥) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)
154, 7, 12, 14syl3anc 1323 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦)
1610eqeq1d 2623 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → (𝑋 = 𝑦 𝐽 = 𝑦))
1716rexbidv 3046 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → (∃𝑦 ∈ (𝒫 𝑥 ∩ Fin)𝑋 = 𝑦 ↔ ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin) 𝐽 = 𝑦))
1815, 17mpbid 222 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ (𝑥𝐽 𝐽 = 𝑥)) → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin) 𝐽 = 𝑦)
1918expr 642 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ 𝑥𝐽) → ( 𝐽 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin) 𝐽 = 𝑦))
203, 19sylan2 491 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) ∧ 𝑥 ∈ 𝒫 𝐽) → ( 𝐽 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin) 𝐽 = 𝑦))
2120ralrimiva 2961 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) → ∀𝑥 ∈ 𝒫 𝐽( 𝐽 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin) 𝐽 = 𝑦))
22 eqid 2621 . . 3 𝐽 = 𝐽
2322iscmp 21114 . 2 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝒫 𝐽( 𝐽 = 𝑥 → ∃𝑦 ∈ (𝒫 𝑥 ∩ Fin) 𝐽 = 𝑦)))
242, 21, 23sylanbrc 697 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ Comp ∧ 𝐽𝐾) → 𝐽 ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  cin 3558  wss 3559  𝒫 cpw 4135   cuni 4407  cfv 5852  Fincfn 7907  Topctop 20630  TopOnctopon 20647  Compccmp 21112
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 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5815  df-fun 5854  df-fv 5860  df-topon 20648  df-cmp 21113
This theorem is referenced by:  kgencmp2  21272  kgen2ss  21281
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