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Theorem ssidcn 12405
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 12392 . . 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 5408 . . . . 5  |-  (  _I  |`  X ) : X -1-1-onto-> X
3 f1of 5370 . . . . 5  |-  ( (  _I  |`  X ) : X -1-1-onto-> X  ->  (  _I  |`  X ) : X --> X )
42, 3ax-mp 5 . . . 4  |-  (  _I  |`  X ) : X --> X
54biantrur 301 . . 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 197 . 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 5200 . . . . . . 7  |-  `' (  _I  |`  X )  =  (  _I  |`  X )
87imaeq1i 4881 . . . . . 6  |-  ( `' (  _I  |`  X )
" x )  =  ( (  _I  |`  X )
" x )
9 elssuni 3767 . . . . . . . . 9  |-  ( x  e.  K  ->  x  C_ 
U. K )
109adantl 275 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  /\  x  e.  K )  ->  x  C_ 
U. K )
11 toponuni 12208 . . . . . . . . 9  |-  ( K  e.  (TopOn `  X
)  ->  X  =  U. K )
1211ad2antlr 480 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  /\  x  e.  K )  ->  X  =  U. K )
1310, 12sseqtrrd 3136 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  /\  x  e.  K )  ->  x  C_  X )
14 resiima 4900 . . . . . . 7  |-  ( x 
C_  X  ->  (
(  _I  |`  X )
" x )  =  x )
1513, 14syl 14 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  /\  x  e.  K )  ->  (
(  _I  |`  X )
" x )  =  x )
168, 15syl5eq 2184 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  /\  x  e.  K )  ->  ( `' (  _I  |`  X )
" x )  =  x )
1716eleq1d 2208 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  /\  x  e.  K )  ->  (
( `' (  _I  |`  X ) " x
)  e.  J  <->  x  e.  J ) )
1817ralbidva 2433 . . 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 3087 . . 3  |-  ( K 
C_  J  <->  A. x  e.  K  x  e.  J )
2018, 19syl6bbr 197 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  ->  ( A. x  e.  K  ( `' (  _I  |`  X )
" x )  e.  J  <->  K  C_  J ) )
216, 20bitrd 187 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    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480   A.wral 2416    C_ wss 3071   U.cuni 3739    _I cid 4213   `'ccnv 4541    |` cres 4544   "cima 4545   -->wf 5122   -1-1-onto->wf1o 5125   ` cfv 5126  (class class class)co 5777  TopOnctopon 12203    Cn ccn 12380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4049  ax-pow 4101  ax-pr 4134  ax-un 4358  ax-setind 4455
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-iun 3818  df-br 3933  df-opab 3993  df-mpt 3994  df-id 4218  df-xp 4548  df-rel 4549  df-cnv 4550  df-co 4551  df-dm 4552  df-rn 4553  df-res 4554  df-ima 4555  df-iota 5091  df-fun 5128  df-fn 5129  df-f 5130  df-f1 5131  df-fo 5132  df-f1o 5133  df-fv 5134  df-ov 5780  df-oprab 5781  df-mpo 5782  df-1st 6041  df-2nd 6042  df-map 6547  df-top 12191  df-topon 12204  df-cn 12383
This theorem is referenced by:  idcn  12407
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