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Theorem sshauslem 17317
Description: Lemma for sshaus 17320 and similar theorems. If the topological property  A is preserved under injective preimages, then a topology finer than one with property  A also has property  A. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypotheses
Ref Expression
t1sep.1  |-  X  = 
U. J
sshauslem.2  |-  ( J  e.  A  ->  J  e.  Top )
sshauslem.3  |-  ( ( J  e.  A  /\  (  _I  |`  X ) : X -1-1-> X  /\  (  _I  |`  X )  e.  ( K  Cn  J ) )  ->  K  e.  A )
Assertion
Ref Expression
sshauslem  |-  ( ( J  e.  A  /\  K  e.  (TopOn `  X
)  /\  J  C_  K
)  ->  K  e.  A )

Proof of Theorem sshauslem
StepHypRef Expression
1 simp1 956 . 2  |-  ( ( J  e.  A  /\  K  e.  (TopOn `  X
)  /\  J  C_  K
)  ->  J  e.  A )
2 f1oi 5617 . . 3  |-  (  _I  |`  X ) : X -1-1-onto-> X
3 f1of1 5577 . . 3  |-  ( (  _I  |`  X ) : X -1-1-onto-> X  ->  (  _I  |`  X ) : X -1-1-> X )
42, 3mp1i 11 . 2  |-  ( ( J  e.  A  /\  K  e.  (TopOn `  X
)  /\  J  C_  K
)  ->  (  _I  |`  X ) : X -1-1-> X )
5 simp3 958 . . 3  |-  ( ( J  e.  A  /\  K  e.  (TopOn `  X
)  /\  J  C_  K
)  ->  J  C_  K
)
6 simp2 957 . . . 4  |-  ( ( J  e.  A  /\  K  e.  (TopOn `  X
)  /\  J  C_  K
)  ->  K  e.  (TopOn `  X ) )
7 sshauslem.2 . . . . . 6  |-  ( J  e.  A  ->  J  e.  Top )
873ad2ant1 977 . . . . 5  |-  ( ( J  e.  A  /\  K  e.  (TopOn `  X
)  /\  J  C_  K
)  ->  J  e.  Top )
9 t1sep.1 . . . . . 6  |-  X  = 
U. J
109toptopon 16888 . . . . 5  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
118, 10sylib 188 . . . 4  |-  ( ( J  e.  A  /\  K  e.  (TopOn `  X
)  /\  J  C_  K
)  ->  J  e.  (TopOn `  X ) )
12 ssidcn 17202 . . . 4  |-  ( ( K  e.  (TopOn `  X )  /\  J  e.  (TopOn `  X )
)  ->  ( (  _I  |`  X )  e.  ( K  Cn  J
)  <->  J  C_  K ) )
136, 11, 12syl2anc 642 . . 3  |-  ( ( J  e.  A  /\  K  e.  (TopOn `  X
)  /\  J  C_  K
)  ->  ( (  _I  |`  X )  e.  ( K  Cn  J
)  <->  J  C_  K ) )
145, 13mpbird 223 . 2  |-  ( ( J  e.  A  /\  K  e.  (TopOn `  X
)  /\  J  C_  K
)  ->  (  _I  |`  X )  e.  ( K  Cn  J ) )
15 sshauslem.3 . 2  |-  ( ( J  e.  A  /\  (  _I  |`  X ) : X -1-1-> X  /\  (  _I  |`  X )  e.  ( K  Cn  J ) )  ->  K  e.  A )
161, 4, 14, 15syl3anc 1183 1  |-  ( ( J  e.  A  /\  K  e.  (TopOn `  X
)  /\  J  C_  K
)  ->  K  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 935    = wceq 1647    e. wcel 1715    C_ wss 3238   U.cuni 3929    _I cid 4407    |` cres 4794   -1-1->wf1 5355   -1-1-onto->wf1o 5357   ` cfv 5358  (class class class)co 5981   Topctop 16848  TopOnctopon 16849    Cn ccn 17171
This theorem is referenced by:  sst0  17318  sst1  17319  sshaus  17320
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-map 6917  df-top 16853  df-topon 16856  df-cn 17174
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