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Theorem hmphen 20576
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 20567 . 2  |-  ( J  ~=  K  <->  ( J Homeo K )  =/=  (/) )
2 n0 3747 . . 3  |-  ( ( J Homeo K )  =/=  (/) 
<->  E. f  f  e.  ( J Homeo K ) )
3 hmeocn 20551 . . . . . 6  |-  ( f  e.  ( J Homeo K )  ->  f  e.  ( J  Cn  K
) )
4 cntop1 20032 . . . . . 6  |-  ( f  e.  ( J  Cn  K )  ->  J  e.  Top )
53, 4syl 17 . . . . 5  |-  ( f  e.  ( J Homeo K )  ->  J  e.  Top )
6 cntop2 20033 . . . . . 6  |-  ( f  e.  ( J  Cn  K )  ->  K  e.  Top )
73, 6syl 17 . . . . 5  |-  ( f  e.  ( J Homeo K )  ->  K  e.  Top )
8 eqid 2402 . . . . . 6  |-  ( x  e.  J  |->  ( f
" x ) )  =  ( x  e.  J  |->  ( f "
x ) )
98hmeoimaf1o 20561 . . . . 5  |-  ( f  e.  ( J Homeo K )  ->  ( x  e.  J  |->  ( f
" x ) ) : J -1-1-onto-> K )
10 f1oen2g 7569 . . . . 5  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  (
x  e.  J  |->  ( f " x ) ) : J -1-1-onto-> K )  ->  J  ~~  K
)
115, 7, 9, 10syl3anc 1230 . . . 4  |-  ( f  e.  ( J Homeo K )  ->  J  ~~  K )
1211exlimiv 1743 . . 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 1633    e. wcel 1842    =/= wne 2598   (/)c0 3737   class class class wbr 4394    |-> cmpt 4452   "cima 4825   -1-1-onto->wf1o 5567  (class class class)co 6277    ~~ cen 7550   Topctop 19684    Cn ccn 20016   Homeochmeo 20544    ~= chmph 20545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-1o 7166  df-map 7458  df-en 7554  df-top 19689  df-topon 19692  df-cn 20019  df-hmeo 20546  df-hmph 20547
This theorem is referenced by:  hmph0  20586  hmphindis  20588
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