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Theorem haushmphlem 17741
Description: Lemma for haushmph 17746 and similar theorems. If the topological property  A is preserved under injective preimages, then property  A is preserved under homeomorphisms. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypotheses
Ref Expression
haushmphlem.1  |-  ( J  e.  A  ->  J  e.  Top )
haushmphlem.2  |-  ( ( J  e.  A  /\  f : U. K -1-1-> U. J  /\  f  e.  ( K  Cn  J ) )  ->  K  e.  A )
Assertion
Ref Expression
haushmphlem  |-  ( J  ~=  K  ->  ( J  e.  A  ->  K  e.  A ) )
Distinct variable groups:    A, f    f, J    f, K

Proof of Theorem haushmphlem
StepHypRef Expression
1 hmphsym 17736 . 2  |-  ( J  ~=  K  ->  K  ~=  J )
2 hmph 17730 . . 3  |-  ( K  ~=  J  <->  ( K  Homeo  J )  =/=  (/) )
3 n0 3581 . . . 4  |-  ( ( K  Homeo  J )  =/=  (/)  <->  E. f  f  e.  ( K  Homeo  J ) )
4 simpl 444 . . . . . . 7  |-  ( ( J  e.  A  /\  f  e.  ( K  Homeo  J ) )  ->  J  e.  A )
5 eqid 2388 . . . . . . . . . 10  |-  U. K  =  U. K
6 eqid 2388 . . . . . . . . . 10  |-  U. J  =  U. J
75, 6hmeof1o 17718 . . . . . . . . 9  |-  ( f  e.  ( K  Homeo  J )  ->  f : U. K -1-1-onto-> U. J )
87adantl 453 . . . . . . . 8  |-  ( ( J  e.  A  /\  f  e.  ( K  Homeo  J ) )  -> 
f : U. K -1-1-onto-> U. J )
9 f1of1 5614 . . . . . . . 8  |-  ( f : U. K -1-1-onto-> U. J  ->  f : U. K -1-1-> U. J )
108, 9syl 16 . . . . . . 7  |-  ( ( J  e.  A  /\  f  e.  ( K  Homeo  J ) )  -> 
f : U. K -1-1-> U. J )
11 hmeocn 17714 . . . . . . . 8  |-  ( f  e.  ( K  Homeo  J )  ->  f  e.  ( K  Cn  J
) )
1211adantl 453 . . . . . . 7  |-  ( ( J  e.  A  /\  f  e.  ( K  Homeo  J ) )  -> 
f  e.  ( K  Cn  J ) )
13 haushmphlem.2 . . . . . . 7  |-  ( ( J  e.  A  /\  f : U. K -1-1-> U. J  /\  f  e.  ( K  Cn  J ) )  ->  K  e.  A )
144, 10, 12, 13syl3anc 1184 . . . . . 6  |-  ( ( J  e.  A  /\  f  e.  ( K  Homeo  J ) )  ->  K  e.  A )
1514expcom 425 . . . . 5  |-  ( f  e.  ( K  Homeo  J )  ->  ( J  e.  A  ->  K  e.  A ) )
1615exlimiv 1641 . . . 4  |-  ( E. f  f  e.  ( K  Homeo  J )  ->  ( J  e.  A  ->  K  e.  A ) )
173, 16sylbi 188 . . 3  |-  ( ( K  Homeo  J )  =/=  (/)  ->  ( J  e.  A  ->  K  e.  A ) )
182, 17sylbi 188 . 2  |-  ( K  ~=  J  ->  ( J  e.  A  ->  K  e.  A ) )
191, 18syl 16 1  |-  ( J  ~=  K  ->  ( J  e.  A  ->  K  e.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   E.wex 1547    e. wcel 1717    =/= wne 2551   (/)c0 3572   U.cuni 3958   class class class wbr 4154   -1-1->wf1 5392   -1-1-onto->wf1o 5394  (class class class)co 6021   Topctop 16882    Cn ccn 17211    Homeo chmeo 17707    ~= chmph 17708
This theorem is referenced by:  t0hmph  17744  t1hmph  17745  haushmph  17746
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-suc 4529  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-1o 6661  df-map 6957  df-top 16887  df-topon 16890  df-cn 17214  df-hmeo 17709  df-hmph 17710
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