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Theorem hmphen 20014
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 20005 . 2  |-  ( J  ~=  K  <->  ( J Homeo K )  =/=  (/) )
2 n0 3787 . . 3  |-  ( ( J Homeo K )  =/=  (/) 
<->  E. f  f  e.  ( J Homeo K ) )
3 hmeocn 19989 . . . . . 6  |-  ( f  e.  ( J Homeo K )  ->  f  e.  ( J  Cn  K
) )
4 cntop1 19500 . . . . . 6  |-  ( f  e.  ( J  Cn  K )  ->  J  e.  Top )
53, 4syl 16 . . . . 5  |-  ( f  e.  ( J Homeo K )  ->  J  e.  Top )
6 cntop2 19501 . . . . . 6  |-  ( f  e.  ( J  Cn  K )  ->  K  e.  Top )
73, 6syl 16 . . . . 5  |-  ( f  e.  ( J Homeo K )  ->  K  e.  Top )
8 eqid 2460 . . . . . 6  |-  ( x  e.  J  |->  ( f
" x ) )  =  ( x  e.  J  |->  ( f "
x ) )
98hmeoimaf1o 19999 . . . . 5  |-  ( f  e.  ( J Homeo K )  ->  ( x  e.  J  |->  ( f
" x ) ) : J -1-1-onto-> K )
10 f1oen2g 7522 . . . . 5  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  (
x  e.  J  |->  ( f " x ) ) : J -1-1-onto-> K )  ->  J  ~~  K
)
115, 7, 9, 10syl3anc 1223 . . . 4  |-  ( f  e.  ( J Homeo K )  ->  J  ~~  K )
1211exlimiv 1693 . . 3  |-  ( E. f  f  e.  ( J Homeo K )  ->  J  ~~  K )
132, 12sylbi 195 . 2  |-  ( ( J Homeo K )  =/=  (/)  ->  J  ~~  K
)
141, 13sylbi 195 1  |-  ( J  ~=  K  ->  J  ~~  K )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   E.wex 1591    e. wcel 1762    =/= wne 2655   (/)c0 3778   class class class wbr 4440    |-> cmpt 4498   "cima 4995   -1-1-onto->wf1o 5578  (class class class)co 6275    ~~ cen 7503   Topctop 19154    Cn ccn 19484   Homeochmeo 19982    ~= chmph 19983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-1o 7120  df-map 7412  df-en 7507  df-top 19159  df-topon 19162  df-cn 19487  df-hmeo 19984  df-hmph 19985
This theorem is referenced by:  hmph0  20024  hmphindis  20026
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