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Theorem sshauslem 17394
Description: Lemma for sshaus 17397 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 957 . 2  |-  ( ( J  e.  A  /\  K  e.  (TopOn `  X
)  /\  J  C_  K
)  ->  J  e.  A )
2 f1oi 5676 . . 3  |-  (  _I  |`  X ) : X -1-1-onto-> X
3 f1of1 5636 . . 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 959 . . 3  |-  ( ( J  e.  A  /\  K  e.  (TopOn `  X
)  /\  J  C_  K
)  ->  J  C_  K
)
6 simp2 958 . . . 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 978 . . . . 5  |-  ( ( J  e.  A  /\  K  e.  (TopOn `  X
)  /\  J  C_  K
)  ->  J  e.  Top )
9 t1sep.1 . . . . . 6  |-  X  = 
U. J
109toptopon 16957 . . . . 5  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
118, 10sylib 189 . . . 4  |-  ( ( J  e.  A  /\  K  e.  (TopOn `  X
)  /\  J  C_  K
)  ->  J  e.  (TopOn `  X ) )
12 ssidcn 17277 . . . 4  |-  ( ( K  e.  (TopOn `  X )  /\  J  e.  (TopOn `  X )
)  ->  ( (  _I  |`  X )  e.  ( K  Cn  J
)  <->  J  C_  K ) )
136, 11, 12syl2anc 643 . . 3  |-  ( ( J  e.  A  /\  K  e.  (TopOn `  X
)  /\  J  C_  K
)  ->  ( (  _I  |`  X )  e.  ( K  Cn  J
)  <->  J  C_  K ) )
145, 13mpbird 224 . 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 1184 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 177    /\ w3a 936    = wceq 1649    e. wcel 1721    C_ wss 3284   U.cuni 3979    _I cid 4457    |` cres 4843   -1-1->wf1 5414   -1-1-onto->wf1o 5416   ` cfv 5417  (class class class)co 6044   Topctop 16917  TopOnctopon 16918    Cn ccn 17246
This theorem is referenced by:  sst0  17395  sst1  17396  sshaus  17397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-map 6983  df-top 16922  df-topon 16925  df-cn 17249
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