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Theorem haushmphlem 17811
Description: Lemma for haushmph 17816 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 17806 . 2  |-  ( J  ~=  K  ->  K  ~=  J )
2 hmph 17800 . . 3  |-  ( K  ~=  J  <->  ( K  Homeo  J )  =/=  (/) )
3 n0 3629 . . . 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 2435 . . . . . . . . . 10  |-  U. K  =  U. K
6 eqid 2435 . . . . . . . . . 10  |-  U. J  =  U. J
75, 6hmeof1o 17788 . . . . . . . . 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 5665 . . . . . . . 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 17784 . . . . . . . 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 1644 . . . 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 1550    e. wcel 1725    =/= wne 2598   (/)c0 3620   U.cuni 4007   class class class wbr 4204   -1-1->wf1 5443   -1-1-onto->wf1o 5445  (class class class)co 6073   Topctop 16950    Cn ccn 17280    Homeo chmeo 17777    ~= chmph 17778
This theorem is referenced by:  t0hmph  17814  t1hmph  17815  haushmph  17816
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-suc 4579  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-1o 6716  df-map 7012  df-top 16955  df-topon 16958  df-cn 17283  df-hmeo 17779  df-hmph 17780
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