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Theorem haushmphlem 17478
Description: Lemma for haushmph 17483 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 17473 . 2  |-  ( J  ~=  K  ->  K  ~=  J )
2 hmph 17467 . . 3  |-  ( K  ~=  J  <->  ( K  Homeo  J )  =/=  (/) )
3 n0 3464 . . . 4  |-  ( ( K  Homeo  J )  =/=  (/)  <->  E. f  f  e.  ( K  Homeo  J ) )
4 simpl 443 . . . . . . 7  |-  ( ( J  e.  A  /\  f  e.  ( K  Homeo  J ) )  ->  J  e.  A )
5 eqid 2283 . . . . . . . . . 10  |-  U. K  =  U. K
6 eqid 2283 . . . . . . . . . 10  |-  U. J  =  U. J
75, 6hmeof1o 17455 . . . . . . . . 9  |-  ( f  e.  ( K  Homeo  J )  ->  f : U. K -1-1-onto-> U. J )
87adantl 452 . . . . . . . 8  |-  ( ( J  e.  A  /\  f  e.  ( K  Homeo  J ) )  -> 
f : U. K -1-1-onto-> U. J )
9 f1of1 5471 . . . . . . . 8  |-  ( f : U. K -1-1-onto-> U. J  ->  f : U. K -1-1-> U. J )
108, 9syl 15 . . . . . . 7  |-  ( ( J  e.  A  /\  f  e.  ( K  Homeo  J ) )  -> 
f : U. K -1-1-> U. J )
11 hmeocn 17451 . . . . . . . 8  |-  ( f  e.  ( K  Homeo  J )  ->  f  e.  ( K  Cn  J
) )
1211adantl 452 . . . . . . 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 1182 . . . . . 6  |-  ( ( J  e.  A  /\  f  e.  ( K  Homeo  J ) )  ->  K  e.  A )
1514expcom 424 . . . . 5  |-  ( f  e.  ( K  Homeo  J )  ->  ( J  e.  A  ->  K  e.  A ) )
1615exlimiv 1666 . . . 4  |-  ( E. f  f  e.  ( K  Homeo  J )  ->  ( J  e.  A  ->  K  e.  A ) )
173, 16sylbi 187 . . 3  |-  ( ( K  Homeo  J )  =/=  (/)  ->  ( J  e.  A  ->  K  e.  A ) )
182, 17sylbi 187 . 2  |-  ( K  ~=  J  ->  ( J  e.  A  ->  K  e.  A ) )
191, 18syl 15 1  |-  ( J  ~=  K  ->  ( J  e.  A  ->  K  e.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   E.wex 1528    e. wcel 1684    =/= wne 2446   (/)c0 3455   U.cuni 3827   class class class wbr 4023   -1-1->wf1 5252   -1-1-onto->wf1o 5254  (class class class)co 5858   Topctop 16631    Cn ccn 16954    Homeo chmeo 17444    ~= chmph 17445
This theorem is referenced by:  t0hmph  17481  t1hmph  17482  haushmph  17483
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-1o 6479  df-map 6774  df-top 16636  df-topon 16639  df-cn 16957  df-hmeo 17446  df-hmph 17447
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