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Theorem ssidcn 16985
Description: The identity function is a continuous function from one topology to another topology on the same set iff the domain is finer than the codomain. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
ssidcn  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  ->  ( (  _I  |`  X )  e.  ( J  Cn  K
)  <->  K  C_  J ) )

Proof of Theorem ssidcn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 iscn 16965 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  ->  ( (  _I  |`  X )  e.  ( J  Cn  K
)  <->  ( (  _I  |`  X ) : X --> X  /\  A. x  e.  K  ( `' (  _I  |`  X ) " x )  e.  J ) ) )
2 f1oi 5511 . . . . 5  |-  (  _I  |`  X ) : X -1-1-onto-> X
3 f1of 5472 . . . . 5  |-  ( (  _I  |`  X ) : X -1-1-onto-> X  ->  (  _I  |`  X ) : X --> X )
42, 3ax-mp 8 . . . 4  |-  (  _I  |`  X ) : X --> X
54biantrur 492 . . 3  |-  ( A. x  e.  K  ( `' (  _I  |`  X )
" x )  e.  J  <->  ( (  _I  |`  X ) : X --> X  /\  A. x  e.  K  ( `' (  _I  |`  X ) " x )  e.  J ) )
61, 5syl6bbr 254 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  ->  ( (  _I  |`  X )  e.  ( J  Cn  K
)  <->  A. x  e.  K  ( `' (  _I  |`  X )
" x )  e.  J ) )
7 cnvresid 5322 . . . . . . 7  |-  `' (  _I  |`  X )  =  (  _I  |`  X )
87imaeq1i 5009 . . . . . 6  |-  ( `' (  _I  |`  X )
" x )  =  ( (  _I  |`  X )
" x )
9 elssuni 3855 . . . . . . . . 9  |-  ( x  e.  K  ->  x  C_ 
U. K )
109adantl 452 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  /\  x  e.  K )  ->  x  C_ 
U. K )
11 toponuni 16665 . . . . . . . . 9  |-  ( K  e.  (TopOn `  X
)  ->  X  =  U. K )
1211ad2antlr 707 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  /\  x  e.  K )  ->  X  =  U. K )
1310, 12sseqtr4d 3215 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  /\  x  e.  K )  ->  x  C_  X )
14 resiima 5029 . . . . . . 7  |-  ( x 
C_  X  ->  (
(  _I  |`  X )
" x )  =  x )
1513, 14syl 15 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  /\  x  e.  K )  ->  (
(  _I  |`  X )
" x )  =  x )
168, 15syl5eq 2327 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  /\  x  e.  K )  ->  ( `' (  _I  |`  X )
" x )  =  x )
1716eleq1d 2349 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  /\  x  e.  K )  ->  (
( `' (  _I  |`  X ) " x
)  e.  J  <->  x  e.  J ) )
1817ralbidva 2559 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  ->  ( A. x  e.  K  ( `' (  _I  |`  X )
" x )  e.  J  <->  A. x  e.  K  x  e.  J )
)
19 dfss3 3170 . . 3  |-  ( K 
C_  J  <->  A. x  e.  K  x  e.  J )
2018, 19syl6bbr 254 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  ->  ( A. x  e.  K  ( `' (  _I  |`  X )
" x )  e.  J  <->  K  C_  J ) )
216, 20bitrd 244 1  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  ->  ( (  _I  |`  X )  e.  ( J  Cn  K
)  <->  K  C_  J ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   U.cuni 3827    _I cid 4304   `'ccnv 4688    |` cres 4691   "cima 4692   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858  TopOnctopon 16632    Cn ccn 16954
This theorem is referenced by:  idcn  16987  sshauslem  17100
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
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