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Theorem sscmp 22779
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 22285 . . 3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝐽 ∈ Top)
213ad2ant1 1134 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Comp ∧ 𝐽 βŠ† 𝐾) β†’ 𝐽 ∈ Top)
3 elpwi 4571 . . . 4 (π‘₯ ∈ 𝒫 𝐽 β†’ π‘₯ βŠ† 𝐽)
4 simpl2 1193 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Comp ∧ 𝐽 βŠ† 𝐾) ∧ (π‘₯ βŠ† 𝐽 ∧ βˆͺ 𝐽 = βˆͺ π‘₯)) β†’ 𝐾 ∈ Comp)
5 simprl 770 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Comp ∧ 𝐽 βŠ† 𝐾) ∧ (π‘₯ βŠ† 𝐽 ∧ βˆͺ 𝐽 = βˆͺ π‘₯)) β†’ π‘₯ βŠ† 𝐽)
6 simpl3 1194 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Comp ∧ 𝐽 βŠ† 𝐾) ∧ (π‘₯ βŠ† 𝐽 ∧ βˆͺ 𝐽 = βˆͺ π‘₯)) β†’ 𝐽 βŠ† 𝐾)
75, 6sstrd 3958 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Comp ∧ 𝐽 βŠ† 𝐾) ∧ (π‘₯ βŠ† 𝐽 ∧ βˆͺ 𝐽 = βˆͺ π‘₯)) β†’ π‘₯ βŠ† 𝐾)
8 simpl1 1192 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Comp ∧ 𝐽 βŠ† 𝐾) ∧ (π‘₯ βŠ† 𝐽 ∧ βˆͺ 𝐽 = βˆͺ π‘₯)) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
9 toponuni 22286 . . . . . . . . 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 772 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Comp ∧ 𝐽 βŠ† 𝐾) ∧ (π‘₯ βŠ† 𝐽 ∧ βˆͺ 𝐽 = βˆͺ π‘₯)) β†’ βˆͺ 𝐽 = βˆͺ π‘₯)
1210, 11eqtrd 2773 . . . . . . 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 22763 . . . . . . 7 ((𝐾 ∈ Comp ∧ π‘₯ βŠ† 𝐾 ∧ 𝑋 = βˆͺ π‘₯) β†’ βˆƒπ‘¦ ∈ (𝒫 π‘₯ ∩ Fin)𝑋 = βˆͺ 𝑦)
154, 7, 12, 14syl3anc 1372 . . . . . 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 2735 . . . . . . 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 3172 . . . . . 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 458 . . . 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 594 . . 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 3140 . 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 2733 . . 3 βˆͺ 𝐽 = βˆͺ 𝐽
2322iscmp 22762 . 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 584 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ Comp ∧ 𝐽 βŠ† 𝐾) β†’ 𝐽 ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070   ∩ cin 3913   βŠ† wss 3914  π’« cpw 4564  βˆͺ cuni 4869  β€˜cfv 6500  Fincfn 8889  Topctop 22265  TopOnctopon 22282  Compccmp 22760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-topon 22283  df-cmp 22761
This theorem is referenced by:  kgencmp2  22920  kgen2ss  22929
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