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Theorem ssidcn 17001
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 16981 . . 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 5527 . . . . 5  |-  (  _I  |`  X ) : X -1-1-onto-> X
3 f1of 5488 . . . . 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 5338 . . . . . . 7  |-  `' (  _I  |`  X )  =  (  _I  |`  X )
87imaeq1i 5025 . . . . . 6  |-  ( `' (  _I  |`  X )
" x )  =  ( (  _I  |`  X )
" x )
9 elssuni 3871 . . . . . . . . 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 16681 . . . . . . . . 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 3228 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  /\  x  e.  K )  ->  x  C_  X )
14 resiima 5045 . . . . . . 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 2340 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  /\  x  e.  K )  ->  ( `' (  _I  |`  X )
" x )  =  x )
1716eleq1d 2362 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  /\  x  e.  K )  ->  (
( `' (  _I  |`  X ) " x
)  e.  J  <->  x  e.  J ) )
1817ralbidva 2572 . . 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 3183 . . 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 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   U.cuni 3843    _I cid 4320   `'ccnv 4704    |` cres 4707   "cima 4708   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874  TopOnctopon 16648    Cn ccn 16970
This theorem is referenced by:  idcn  17003  sshauslem  17116
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-top 16652  df-topon 16655  df-cn 16973
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