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Theorem sshauslem 17441
Description: Lemma for sshaus 17444 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 958 . 2  |-  ( ( J  e.  A  /\  K  e.  (TopOn `  X
)  /\  J  C_  K
)  ->  J  e.  A )
2 f1oi 5716 . . 3  |-  (  _I  |`  X ) : X -1-1-onto-> X
3 f1of1 5676 . . 3  |-  ( (  _I  |`  X ) : X -1-1-onto-> X  ->  (  _I  |`  X ) : X -1-1-> X )
42, 3mp1i 12 . 2  |-  ( ( J  e.  A  /\  K  e.  (TopOn `  X
)  /\  J  C_  K
)  ->  (  _I  |`  X ) : X -1-1-> X )
5 simp3 960 . . 3  |-  ( ( J  e.  A  /\  K  e.  (TopOn `  X
)  /\  J  C_  K
)  ->  J  C_  K
)
6 simp2 959 . . . 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 979 . . . . 5  |-  ( ( J  e.  A  /\  K  e.  (TopOn `  X
)  /\  J  C_  K
)  ->  J  e.  Top )
9 t1sep.1 . . . . . 6  |-  X  = 
U. J
109toptopon 17003 . . . . 5  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
118, 10sylib 190 . . . 4  |-  ( ( J  e.  A  /\  K  e.  (TopOn `  X
)  /\  J  C_  K
)  ->  J  e.  (TopOn `  X ) )
12 ssidcn 17324 . . . 4  |-  ( ( K  e.  (TopOn `  X )  /\  J  e.  (TopOn `  X )
)  ->  ( (  _I  |`  X )  e.  ( K  Cn  J
)  <->  J  C_  K ) )
136, 11, 12syl2anc 644 . . 3  |-  ( ( J  e.  A  /\  K  e.  (TopOn `  X
)  /\  J  C_  K
)  ->  ( (  _I  |`  X )  e.  ( K  Cn  J
)  <->  J  C_  K ) )
145, 13mpbird 225 . 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 1185 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 178    /\ w3a 937    = wceq 1653    e. wcel 1726    C_ wss 3322   U.cuni 4017    _I cid 4496    |` cres 4883   -1-1->wf1 5454   -1-1-onto->wf1o 5456   ` cfv 5457  (class class class)co 6084   Topctop 16963  TopOnctopon 16964    Cn ccn 17293
This theorem is referenced by:  sst0  17442  sst1  17443  sshaus  17444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-map 7023  df-top 16968  df-topon 16971  df-cn 17296
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