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Theorem ssidcn 17321
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 17301 . . 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 5715 . . . . 5  |-  (  _I  |`  X ) : X -1-1-onto-> X
3 f1of 5676 . . . . 5  |-  ( (  _I  |`  X ) : X -1-1-onto-> X  ->  (  _I  |`  X ) : X --> X )
42, 3ax-mp 8 . . . 4  |-  (  _I  |`  X ) : X --> X
54biantrur 494 . . 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 256 . 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 5525 . . . . . . 7  |-  `' (  _I  |`  X )  =  (  _I  |`  X )
87imaeq1i 5202 . . . . . 6  |-  ( `' (  _I  |`  X )
" x )  =  ( (  _I  |`  X )
" x )
9 elssuni 4045 . . . . . . . . 9  |-  ( x  e.  K  ->  x  C_ 
U. K )
109adantl 454 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  /\  x  e.  K )  ->  x  C_ 
U. K )
11 toponuni 16994 . . . . . . . . 9  |-  ( K  e.  (TopOn `  X
)  ->  X  =  U. K )
1211ad2antlr 709 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  /\  x  e.  K )  ->  X  =  U. K )
1310, 12sseqtr4d 3387 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  /\  x  e.  K )  ->  x  C_  X )
14 resiima 5222 . . . . . . 7  |-  ( x 
C_  X  ->  (
(  _I  |`  X )
" x )  =  x )
1513, 14syl 16 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  /\  x  e.  K )  ->  (
(  _I  |`  X )
" x )  =  x )
168, 15syl5eq 2482 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  /\  x  e.  K )  ->  ( `' (  _I  |`  X )
" x )  =  x )
1716eleq1d 2504 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  /\  x  e.  K )  ->  (
( `' (  _I  |`  X ) " x
)  e.  J  <->  x  e.  J ) )
1817ralbidva 2723 . . 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 3340 . . 3  |-  ( K 
C_  J  <->  A. x  e.  K  x  e.  J )
2018, 19syl6bbr 256 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  ->  ( A. x  e.  K  ( `' (  _I  |`  X )
" x )  e.  J  <->  K  C_  J ) )
216, 20bitrd 246 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707    C_ wss 3322   U.cuni 4017    _I cid 4495   `'ccnv 4879    |` cres 4882   "cima 4883   -->wf 5452   -1-1-onto->wf1o 5455   ` cfv 5456  (class class class)co 6083  TopOnctopon 16961    Cn ccn 17290
This theorem is referenced by:  idcn  17323  sshauslem  17438
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-map 7022  df-top 16965  df-topon 16968  df-cn 17293
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