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Theorem hmphen 17476
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 17467 . 2  |-  ( J  ~=  K  <->  ( J  Homeo  K )  =/=  (/) )
2 n0 3464 . . 3  |-  ( ( J  Homeo  K )  =/=  (/)  <->  E. f  f  e.  ( J  Homeo  K ) )
3 hmeocn 17451 . . . . . 6  |-  ( f  e.  ( J  Homeo  K )  ->  f  e.  ( J  Cn  K
) )
4 cntop1 16970 . . . . . 6  |-  ( f  e.  ( J  Cn  K )  ->  J  e.  Top )
53, 4syl 15 . . . . 5  |-  ( f  e.  ( J  Homeo  K )  ->  J  e.  Top )
6 cntop2 16971 . . . . . 6  |-  ( f  e.  ( J  Cn  K )  ->  K  e.  Top )
73, 6syl 15 . . . . 5  |-  ( f  e.  ( J  Homeo  K )  ->  K  e.  Top )
8 eqid 2283 . . . . . 6  |-  ( x  e.  J  |->  ( f
" x ) )  =  ( x  e.  J  |->  ( f "
x ) )
98hmeoimaf1o 17461 . . . . 5  |-  ( f  e.  ( J  Homeo  K )  ->  ( x  e.  J  |->  ( f
" x ) ) : J -1-1-onto-> K )
10 f1oen2g 6878 . . . . 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 1666 . . 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 1528    e. wcel 1684    =/= wne 2446   (/)c0 3455   class class class wbr 4023    e. cmpt 4077   "cima 4692   -1-1-onto->wf1o 5254  (class class class)co 5858    ~~ cen 6860   Topctop 16631    Cn ccn 16954    Homeo chmeo 17444    ~= chmph 17445
This theorem is referenced by:  hmph0  17486  hmphindis  17488
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-1o 6479  df-map 6774  df-en 6864  df-top 16636  df-topon 16639  df-cn 16957  df-hmeo 17446  df-hmph 17447
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