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Theorem hmphen2 17546
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 17523 . 2  |-  ( J  ~=  K  <->  ( J  Homeo  K )  =/=  (/) )
2 n0 3498 . . 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 17511 . . . . 5  |-  ( f  e.  ( J  Homeo  K )  ->  f : X
-1-1-onto-> Y )
6 f1oen3g 6920 . . . . 5  |-  ( ( f  e.  ( J 
Homeo  K )  /\  f : X -1-1-onto-> Y )  ->  X  ~~  Y )
75, 6mpdan 649 . . . 4  |-  ( f  e.  ( J  Homeo  K )  ->  X  ~~  Y )
87exlimiv 1625 . . 3  |-  ( E. f  f  e.  ( J  Homeo  K )  ->  X  ~~  Y )
92, 8sylbi 187 . 2  |-  ( ( J  Homeo  K )  =/=  (/)  ->  X  ~~  Y )
101, 9sylbi 187 1  |-  ( J  ~=  K  ->  X  ~~  Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1532    = wceq 1633    e. wcel 1701    =/= wne 2479   (/)c0 3489   U.cuni 3864   class class class wbr 4060   -1-1-onto->wf1o 5291  (class class class)co 5900    ~~ cen 6903    Homeo chmeo 17500    ~= chmph 17501
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-suc 4435  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-1o 6521  df-map 6817  df-en 6907  df-top 16692  df-topon 16695  df-cn 17013  df-hmeo 17502  df-hmph 17503
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