Theorem ssidcn 21059

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.

Step	Hyp	Ref	Expression
1		iscn 21039	. . 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 6174	. . . . 5 |- ( _I |` X ) : X -1-1-onto-> X
3		f1of 6137	. . . . 5 |- ( ( _I |` X ) : X -1-1-onto-> X -> ( _I |` X ) : X --> X )
4	2, 3	ax-mp 5	. . . 4 |- ( _I |` X ) : X --> X
5	4	biantrur 527	. . 3 |- ( A. x e. K ( `' ( _I |` X ) " x ) e. J <-> ( ( _I |` X ) : X --> X /\ A. x e. K ( `' ( _I |` X ) " x ) e. J ) )
6	1, 5	syl6bbr 278	. 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 5968	. . . . . . 7 |- `' ( _I |` X ) = ( _I |` X )
8	7	imaeq1i 5463	. . . . . 6 |- ( `' ( _I |` X ) " x ) = ( ( _I |` X ) " x )
9		elssuni 4467	. . . . . . . . 9 |- ( x e. K -> x C_ U. K )
10	9	adantl 482	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ x e. K ) -> x C_ U. K )
11		toponuni 20719	. . . . . . . . 9 |- ( K e. (TopOn ` X ) -> X = U. K )
12	11	ad2antlr 763	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ x e. K ) -> X = U. K )
13	10, 12	sseqtr4d 3642	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ x e. K ) -> x C_ X )
14		resiima 5480	. . . . . . 7 |- ( x C_ X -> ( ( _I |` X ) " x ) = x )
15	13, 14	syl 17	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ x e. K ) -> ( ( _I |` X ) " x ) = x )
16	8, 15	syl5eq 2668	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ x e. K ) -> ( `' ( _I |` X ) " x ) = x )
17	16	eleq1d 2686	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ x e. K ) -> ( ( `' ( _I |` X ) " x ) e. J <-> x e. J ) )
18	17	ralbidva 2985	. . 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 3592	. . 3 |- ( K C_ J <-> A. x e. K x e. J )
20	18, 19	syl6bbr 278	. 2 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) -> ( A. x e. K ( `' ( _I |` X ) " x ) e. J <-> K C_ J ) )
21	6, 20	bitrd 268	1 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) -> ( ( _I |` X ) e. ( J Cn K ) <-> K C_ J ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   U.cuni 4436   _I cid 5023   `'ccnv 5113   |` cres 5116   "cima 5117   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650  TopOnctopon 20715   Cn ccn 21028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-top 20699  df-topon 20716  df-cn 21031

This theorem is referenced by:  idcn  21061  sshauslem  21176