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Theorem sscmp 22908
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 = βˆͺ 𝐾
Assertion
Ref Expression
sscmp ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Comp ∧ 𝐽 βŠ† 𝐾) β†’ 𝐽 ∈ Comp)
Proof of Theorem sscmp
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 topontop 22414 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
213ad2ant1 1133 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Comp ∧ 𝐽 βŠ† 𝐾) β†’ 𝐽 ∈ Top)
3 elpwi 4609 . . . 4 (π‘₯ ∈ 𝒫 𝐽 β†’ π‘₯ βŠ† 𝐽)
4 simpl2 1192 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Comp ∧ 𝐽 βŠ† 𝐾) ∧ (π‘₯ βŠ† 𝐽 ∧ βˆͺ 𝐽 = βˆͺ π‘₯)) β†’ 𝐾 ∈ Comp)
5 simprl 769 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Comp ∧ 𝐽 βŠ† 𝐾) ∧ (π‘₯ βŠ† 𝐽 ∧ βˆͺ 𝐽 = βˆͺ π‘₯)) β†’ π‘₯ βŠ† 𝐽)
6 simpl3 1193 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Comp ∧ 𝐽 βŠ† 𝐾) ∧ (π‘₯ βŠ† 𝐽 ∧ βˆͺ 𝐽 = βˆͺ π‘₯)) β†’ 𝐽 βŠ† 𝐾)
75, 6sstrd 3992 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Comp ∧ 𝐽 βŠ† 𝐾) ∧ (π‘₯ βŠ† 𝐽 ∧ βˆͺ 𝐽 = βˆͺ π‘₯)) β†’ π‘₯ βŠ† 𝐾)
8 simpl1 1191 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Comp ∧ 𝐽 βŠ† 𝐾) ∧ (π‘₯ βŠ† 𝐽 ∧ βˆͺ 𝐽 = βˆͺ π‘₯)) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
9 toponuni 22415 . . . . . . . . 9 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
108, 9syl 17 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Comp ∧ 𝐽 βŠ† 𝐾) ∧ (π‘₯ βŠ† 𝐽 ∧ βˆͺ 𝐽 = βˆͺ π‘₯)) β†’ 𝑋 = βˆͺ 𝐽)
11 simprr 771 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Comp ∧ 𝐽 βŠ† 𝐾) ∧ (π‘₯ βŠ† 𝐽 ∧ βˆͺ 𝐽 = βˆͺ π‘₯)) β†’ βˆͺ 𝐽 = βˆͺ π‘₯)
1210, 11eqtrd 2772 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Comp ∧ 𝐽 βŠ† 𝐾) ∧ (π‘₯ βŠ† 𝐽 ∧ βˆͺ 𝐽 = βˆͺ π‘₯)) β†’ 𝑋 = βˆͺ π‘₯)
13 sscmp.1 . . . . . . . 8 𝑋 = βˆͺ 𝐾
1413cmpcov 22892 . . . . . . 7 ((𝐾 ∈ Comp ∧ π‘₯ βŠ† 𝐾 ∧ 𝑋 = βˆͺ π‘₯) β†’ βˆƒπ‘¦ ∈ (𝒫 π‘₯ ∩ Fin)𝑋 = βˆͺ 𝑦)
154, 7, 12, 14syl3anc 1371 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Comp ∧ 𝐽 βŠ† 𝐾) ∧ (π‘₯ βŠ† 𝐽 ∧ βˆͺ 𝐽 = βˆͺ π‘₯)) β†’ βˆƒπ‘¦ ∈ (𝒫 π‘₯ ∩ Fin)𝑋 = βˆͺ 𝑦)
1610eqeq1d 2734 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Comp ∧ 𝐽 βŠ† 𝐾) ∧ (π‘₯ βŠ† 𝐽 ∧ βˆͺ 𝐽 = βˆͺ π‘₯)) β†’ (𝑋 = βˆͺ 𝑦 ↔ βˆͺ 𝐽 = βˆͺ 𝑦))
1716rexbidv 3178 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Comp ∧ 𝐽 βŠ† 𝐾) ∧ (π‘₯ βŠ† 𝐽 ∧ βˆͺ 𝐽 = βˆͺ π‘₯)) β†’ (βˆƒπ‘¦ ∈ (𝒫 π‘₯ ∩ Fin)𝑋 = βˆͺ 𝑦 ↔ βˆƒπ‘¦ ∈ (𝒫 π‘₯ ∩ Fin)βˆͺ 𝐽 = βˆͺ 𝑦))
1815, 17mpbid 231 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Comp ∧ 𝐽 βŠ† 𝐾) ∧ (π‘₯ βŠ† 𝐽 ∧ βˆͺ 𝐽 = βˆͺ π‘₯)) β†’ βˆƒπ‘¦ ∈ (𝒫 π‘₯ ∩ Fin)βˆͺ 𝐽 = βˆͺ 𝑦)
1918expr 457 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Comp ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ βŠ† 𝐽) β†’ (βˆͺ 𝐽 = βˆͺ π‘₯ β†’ βˆƒπ‘¦ ∈ (𝒫 π‘₯ ∩ Fin)βˆͺ 𝐽 = βˆͺ 𝑦))
203, 19sylan2 593 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Comp ∧ 𝐽 βŠ† 𝐾) ∧ π‘₯ ∈ 𝒫 𝐽) β†’ (βˆͺ 𝐽 = βˆͺ π‘₯ β†’ βˆƒπ‘¦ ∈ (𝒫 π‘₯ ∩ Fin)βˆͺ 𝐽 = βˆͺ 𝑦))
2120ralrimiva 3146 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Comp ∧ 𝐽 βŠ† 𝐾) β†’ βˆ€π‘₯ ∈ 𝒫 𝐽(βˆͺ 𝐽 = βˆͺ π‘₯ β†’ βˆƒπ‘¦ ∈ (𝒫 π‘₯ ∩ Fin)βˆͺ 𝐽 = βˆͺ 𝑦))
22 eqid 2732 . . 3 βˆͺ 𝐽 = βˆͺ 𝐽
2322iscmp 22891 . 2 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ βˆ€π‘₯ ∈ 𝒫 𝐽(βˆͺ 𝐽 = βˆͺ π‘₯ β†’ βˆƒπ‘¦ ∈ (𝒫 π‘₯ ∩ Fin)βˆͺ 𝐽 = βˆͺ 𝑦)))
242, 21, 23sylanbrc 583 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Comp ∧ 𝐽 βŠ† 𝐾) β†’ 𝐽 ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070   ∩ cin 3947   βŠ† wss 3948  π’« cpw 4602  βˆͺ cuni 4908  β€˜cfv 6543  Fincfn 8938  Topctop 22394  TopOnctopon 22411  Compccmp 22889
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-topon 22412  df-cmp 22890
This theorem is referenced by:  kgencmp2  23049  kgen2ss  23058
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