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Theorem sshauslem 20388
Description: Lemma for sshaus 20391 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 1008 . 2  |-  ( ( J  e.  A  /\  K  e.  (TopOn `  X
)  /\  J  C_  K
)  ->  J  e.  A )
2 f1oi 5850 . . 3  |-  (  _I  |`  X ) : X -1-1-onto-> X
3 f1of1 5813 . . 3  |-  ( (  _I  |`  X ) : X -1-1-onto-> X  ->  (  _I  |`  X ) : X -1-1-> X )
42, 3mp1i 13 . 2  |-  ( ( J  e.  A  /\  K  e.  (TopOn `  X
)  /\  J  C_  K
)  ->  (  _I  |`  X ) : X -1-1-> X )
5 simp3 1010 . . 3  |-  ( ( J  e.  A  /\  K  e.  (TopOn `  X
)  /\  J  C_  K
)  ->  J  C_  K
)
6 simp2 1009 . . . 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 1029 . . . . 5  |-  ( ( J  e.  A  /\  K  e.  (TopOn `  X
)  /\  J  C_  K
)  ->  J  e.  Top )
9 t1sep.1 . . . . . 6  |-  X  = 
U. J
109toptopon 19948 . . . . 5  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
118, 10sylib 200 . . . 4  |-  ( ( J  e.  A  /\  K  e.  (TopOn `  X
)  /\  J  C_  K
)  ->  J  e.  (TopOn `  X ) )
12 ssidcn 20271 . . . 4  |-  ( ( K  e.  (TopOn `  X )  /\  J  e.  (TopOn `  X )
)  ->  ( (  _I  |`  X )  e.  ( K  Cn  J
)  <->  J  C_  K ) )
136, 11, 12syl2anc 667 . . 3  |-  ( ( J  e.  A  /\  K  e.  (TopOn `  X
)  /\  J  C_  K
)  ->  ( (  _I  |`  X )  e.  ( K  Cn  J
)  <->  J  C_  K ) )
145, 13mpbird 236 . 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 1268 1  |-  ( ( J  e.  A  /\  K  e.  (TopOn `  X
)  /\  J  C_  K
)  ->  K  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ w3a 985    = wceq 1444    e. wcel 1887    C_ wss 3404   U.cuni 4198    _I cid 4744    |` cres 4836   -1-1->wf1 5579   -1-1-onto->wf1o 5581   ` cfv 5582  (class class class)co 6290   Topctop 19917  TopOnctopon 19918    Cn ccn 20240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-map 7474  df-top 19921  df-topon 19923  df-cn 20243
This theorem is referenced by:  sst0  20389  sst1  20390  sshaus  20391
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