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Theorem hmphen2 17831
Description: Homeomorphisms preserve the cardinality of the underlying sets. (Contributed by FL, 17-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
Hypotheses
Ref Expression
cmphaushmeo.1  |-  X  = 
U. J
cmphaushmeo.2  |-  Y  = 
U. K
Assertion
Ref Expression
hmphen2  |-  ( J  ~=  K  ->  X  ~~  Y )

Proof of Theorem hmphen2
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 hmph 17808 . 2  |-  ( J  ~=  K  <->  ( J  Homeo  K )  =/=  (/) )
2 n0 3637 . . 3  |-  ( ( J  Homeo  K )  =/=  (/)  <->  E. f  f  e.  ( J  Homeo  K ) )
3 cmphaushmeo.1 . . . . . 6  |-  X  = 
U. J
4 cmphaushmeo.2 . . . . . 6  |-  Y  = 
U. K
53, 4hmeof1o 17796 . . . . 5  |-  ( f  e.  ( J  Homeo  K )  ->  f : X
-1-1-onto-> Y )
6 f1oen3g 7123 . . . . 5  |-  ( ( f  e.  ( J 
Homeo  K )  /\  f : X -1-1-onto-> Y )  ->  X  ~~  Y )
75, 6mpdan 650 . . . 4  |-  ( f  e.  ( J  Homeo  K )  ->  X  ~~  Y )
87exlimiv 1644 . . 3  |-  ( E. f  f  e.  ( J  Homeo  K )  ->  X  ~~  Y )
92, 8sylbi 188 . 2  |-  ( ( J  Homeo  K )  =/=  (/)  ->  X  ~~  Y )
101, 9sylbi 188 1  |-  ( J  ~=  K  ->  X  ~~  Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2599   (/)c0 3628   U.cuni 4015   class class class wbr 4212   -1-1-onto->wf1o 5453  (class class class)co 6081    ~~ cen 7106    Homeo chmeo 17785    ~= chmph 17786
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-1o 6724  df-map 7020  df-en 7110  df-top 16963  df-topon 16966  df-cn 17291  df-hmeo 17787  df-hmph 17788
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