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Theorem ssidcn 19515
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 19495 . . 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 5842 . . . . 5  |-  (  _I  |`  X ) : X -1-1-onto-> X
3 f1of 5807 . . . . 5  |-  ( (  _I  |`  X ) : X -1-1-onto-> X  ->  (  _I  |`  X ) : X --> X )
42, 3ax-mp 5 . . . 4  |-  (  _I  |`  X ) : X --> X
54biantrur 506 . . 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 263 . 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 5649 . . . . . . 7  |-  `' (  _I  |`  X )  =  (  _I  |`  X )
87imaeq1i 5325 . . . . . 6  |-  ( `' (  _I  |`  X )
" x )  =  ( (  _I  |`  X )
" x )
9 elssuni 4268 . . . . . . . . 9  |-  ( x  e.  K  ->  x  C_ 
U. K )
109adantl 466 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  /\  x  e.  K )  ->  x  C_ 
U. K )
11 toponuni 19188 . . . . . . . . 9  |-  ( K  e.  (TopOn `  X
)  ->  X  =  U. K )
1211ad2antlr 726 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  /\  x  e.  K )  ->  X  =  U. K )
1310, 12sseqtr4d 3534 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  /\  x  e.  K )  ->  x  C_  X )
14 resiima 5342 . . . . . . 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 2513 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  /\  x  e.  K )  ->  ( `' (  _I  |`  X )
" x )  =  x )
1716eleq1d 2529 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  /\  x  e.  K )  ->  (
( `' (  _I  |`  X ) " x
)  e.  J  <->  x  e.  J ) )
1817ralbidva 2893 . . 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 3487 . . 3  |-  ( K 
C_  J  <->  A. x  e.  K  x  e.  J )
2018, 19syl6bbr 263 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  ->  ( A. x  e.  K  ( `' (  _I  |`  X )
" x )  e.  J  <->  K  C_  J ) )
216, 20bitrd 253 1  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  X )
)  ->  ( (  _I  |`  X )  e.  ( J  Cn  K
)  <->  K  C_  J ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2807    C_ wss 3469   U.cuni 4238    _I cid 4783   `'ccnv 4991    |` cres 4994   "cima 4995   -->wf 5575   -1-1-onto->wf1o 5578   ` cfv 5579  (class class class)co 6275  TopOnctopon 19155    Cn ccn 19484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-map 7412  df-top 19159  df-topon 19162  df-cn 19487
This theorem is referenced by:  idcn  19517  sshauslem  19632
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