Theorem ssidcn 14884

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 14871	. . 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 5611	. . . . 5 |- ( _I |` X ) : X -1-1-onto-> X
3		f1of 5572	. . . . 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 303	. . 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	bitr4di 198	. 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 5395	. . . . . . 7 |- `' ( _I |` X ) = ( _I |` X )
8	7	imaeq1i 5065	. . . . . 6 |- ( `' ( _I |` X ) " x ) = ( ( _I |` X ) " x )
9		elssuni 3916	. . . . . . . . 9 |- ( x e. K -> x C_ U. K )
10	9	adantl 277	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ x e. K ) -> x C_ U. K )
11		toponuni 14689	. . . . . . . . 9 |- ( K e. (TopOn ` X ) -> X = U. K )
12	11	ad2antlr 489	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ x e. K ) -> X = U. K )
13	10, 12	sseqtrrd 3263	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ x e. K ) -> x C_ X )
14		resiima 5086	. . . . . . 7 |- ( x C_ X -> ( ( _I |` X ) " x ) = x )
15	13, 14	syl 14	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ x e. K ) -> ( ( _I |` X ) " x ) = x )
16	8, 15	eqtrid 2274	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ x e. K ) -> ( `' ( _I |` X ) " x ) = x )
17	16	eleq1d 2298	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ x e. K ) -> ( ( `' ( _I |` X ) " x ) e. J <-> x e. J ) )
18	17	ralbidva 2526	. . 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 3213	. . 3 |- ( K C_ J <-> A. x e. K x e. J )
20	18, 19	bitr4di 198	. 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 188	1 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) -> ( ( _I |` X ) e. ( J Cn K ) <-> K C_ J ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   A.wral 2508   C_ wss 3197   U.cuni 3888   _I cid 4379   `'ccnv 4718   |` cres 4721   "cima 4722   -->wf 5314   -1-1-onto->wf1o 5317   ` cfv 5318  (class class class)co 6001  TopOnctopon 14684   Cn ccn 14859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-map 6797  df-top 14672  df-topon 14685  df-cn 14862

This theorem is referenced by:  idcn  14886