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Theorem cmphmph 17821
Description: Compactness is a topological property-that is, for any two homeomorphic topologies, either both are compact or neither is. (Contributed by Jeff Hankins, 30-Jun-2009.) (Revised by Mario Carneiro, 23-Aug-2015.)
Assertion
Ref Expression
cmphmph  |-  ( J  ~=  K  ->  ( J  e.  Comp  ->  K  e.  Comp ) )

Proof of Theorem cmphmph
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 hmph 17809 . 2  |-  ( J  ~=  K  <->  ( J  Homeo  K )  =/=  (/) )
2 n0 3638 . . 3  |-  ( ( J  Homeo  K )  =/=  (/)  <->  E. f  f  e.  ( J  Homeo  K ) )
3 eqid 2437 . . . . . . 7  |-  U. J  =  U. J
4 eqid 2437 . . . . . . 7  |-  U. K  =  U. K
53, 4hmeof1o 17797 . . . . . 6  |-  ( f  e.  ( J  Homeo  K )  ->  f : U. J -1-1-onto-> U. K )
6 f1ofo 5682 . . . . . 6  |-  ( f : U. J -1-1-onto-> U. K  ->  f : U. J -onto-> U. K )
75, 6syl 16 . . . . 5  |-  ( f  e.  ( J  Homeo  K )  ->  f : U. J -onto-> U. K )
8 hmeocn 17793 . . . . 5  |-  ( f  e.  ( J  Homeo  K )  ->  f  e.  ( J  Cn  K
) )
94cncmp 17456 . . . . . . 7  |-  ( ( J  e.  Comp  /\  f : U. J -onto-> U. K  /\  f  e.  ( J  Cn  K ) )  ->  K  e.  Comp )
1093expb 1155 . . . . . 6  |-  ( ( J  e.  Comp  /\  (
f : U. J -onto-> U. K  /\  f  e.  ( J  Cn  K
) ) )  ->  K  e.  Comp )
1110expcom 426 . . . . 5  |-  ( ( f : U. J -onto-> U. K  /\  f  e.  ( J  Cn  K
) )  ->  ( J  e.  Comp  ->  K  e.  Comp ) )
127, 8, 11syl2anc 644 . . . 4  |-  ( f  e.  ( J  Homeo  K )  ->  ( J  e.  Comp  ->  K  e.  Comp ) )
1312exlimiv 1645 . . 3  |-  ( E. f  f  e.  ( J  Homeo  K )  ->  ( J  e.  Comp  ->  K  e.  Comp ) )
142, 13sylbi 189 . 2  |-  ( ( J  Homeo  K )  =/=  (/)  ->  ( J  e.  Comp  ->  K  e.  Comp ) )
151, 14sylbi 189 1  |-  ( J  ~=  K  ->  ( J  e.  Comp  ->  K  e.  Comp ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   E.wex 1551    e. wcel 1726    =/= wne 2600   (/)c0 3629   U.cuni 4016   class class class wbr 4213   -onto->wfo 5453   -1-1-onto->wf1o 5454  (class class class)co 6082    Cn ccn 17289   Compccmp 17450    Homeo chmeo 17786    ~= chmph 17787
This theorem is referenced by:  ptcmpfi  17846  xrcmp  18974  reheibor  26549
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-recs 6634  df-rdg 6669  df-1o 6725  df-oadd 6729  df-er 6906  df-map 7021  df-en 7111  df-dom 7112  df-fin 7114  df-top 16964  df-topon 16967  df-cn 17292  df-cmp 17451  df-hmeo 17788  df-hmph 17789
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