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Theorem hmphen 17492
Description: Homeomorphisms preserve the cardinality of the topologies. (Contributed by FL, 1-Jun-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
hmphen  |-  ( J  ~=  K  ->  J  ~~  K )

Proof of Theorem hmphen
Dummy variables  x  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hmph 17483 . 2  |-  ( J  ~=  K  <->  ( J  Homeo  K )  =/=  (/) )
2 n0 3477 . . 3  |-  ( ( J  Homeo  K )  =/=  (/)  <->  E. f  f  e.  ( J  Homeo  K ) )
3 hmeocn 17467 . . . . . 6  |-  ( f  e.  ( J  Homeo  K )  ->  f  e.  ( J  Cn  K
) )
4 cntop1 16986 . . . . . 6  |-  ( f  e.  ( J  Cn  K )  ->  J  e.  Top )
53, 4syl 15 . . . . 5  |-  ( f  e.  ( J  Homeo  K )  ->  J  e.  Top )
6 cntop2 16987 . . . . . 6  |-  ( f  e.  ( J  Cn  K )  ->  K  e.  Top )
73, 6syl 15 . . . . 5  |-  ( f  e.  ( J  Homeo  K )  ->  K  e.  Top )
8 eqid 2296 . . . . . 6  |-  ( x  e.  J  |->  ( f
" x ) )  =  ( x  e.  J  |->  ( f "
x ) )
98hmeoimaf1o 17477 . . . . 5  |-  ( f  e.  ( J  Homeo  K )  ->  ( x  e.  J  |->  ( f
" x ) ) : J -1-1-onto-> K )
10 f1oen2g 6894 . . . . 5  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  (
x  e.  J  |->  ( f " x ) ) : J -1-1-onto-> K )  ->  J  ~~  K
)
115, 7, 9, 10syl3anc 1182 . . . 4  |-  ( f  e.  ( J  Homeo  K )  ->  J  ~~  K )
1211exlimiv 1624 . . 3  |-  ( E. f  f  e.  ( J  Homeo  K )  ->  J  ~~  K )
132, 12sylbi 187 . 2  |-  ( ( J  Homeo  K )  =/=  (/)  ->  J  ~~  K )
141, 13sylbi 187 1  |-  ( J  ~=  K  ->  J  ~~  K )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1531    e. wcel 1696    =/= wne 2459   (/)c0 3468   class class class wbr 4039    e. cmpt 4093   "cima 4708   -1-1-onto->wf1o 5270  (class class class)co 5874    ~~ cen 6876   Topctop 16647    Cn ccn 16970    Homeo chmeo 17460    ~= chmph 17461
This theorem is referenced by:  hmph0  17502  hmphindis  17504
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-1o 6495  df-map 6790  df-en 6880  df-top 16652  df-topon 16655  df-cn 16973  df-hmeo 17462  df-hmph 17463
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