MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sshauslem Unicode version

Theorem sshauslem 17100
Description: Lemma for sshaus 17103 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 955 . 2  |-  ( ( J  e.  A  /\  K  e.  (TopOn `  X
)  /\  J  C_  K
)  ->  J  e.  A )
2 f1oi 5511 . . 3  |-  (  _I  |`  X ) : X -1-1-onto-> X
3 f1of1 5471 . . 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 957 . . 3  |-  ( ( J  e.  A  /\  K  e.  (TopOn `  X
)  /\  J  C_  K
)  ->  J  C_  K
)
6 simp2 956 . . . 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 976 . . . . 5  |-  ( ( J  e.  A  /\  K  e.  (TopOn `  X
)  /\  J  C_  K
)  ->  J  e.  Top )
9 t1sep.1 . . . . . 6  |-  X  = 
U. J
109toptopon 16671 . . . . 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 16985 . . . 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 1182 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 934    = wceq 1623    e. wcel 1684    C_ wss 3152   U.cuni 3827    _I cid 4304    |` cres 4691   -1-1->wf1 5252   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   Topctop 16631  TopOnctopon 16632    Cn ccn 16954
This theorem is referenced by:  sst0  17101  sst1  17102  sshaus  17103
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-top 16636  df-topon 16639  df-cn 16957
  Copyright terms: Public domain W3C validator