Theorem ssidcn 13281

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 13268	. . 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 5491	. . . . 5 |- ( _I |` X ) : X -1-1-onto-> X
3		f1of 5453	. . . . 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 5282	. . . . . . 7 |- `' ( _I |` X ) = ( _I |` X )
8	7	imaeq1i 4960	. . . . . 6 |- ( `' ( _I |` X ) " x ) = ( ( _I |` X ) " x )
9		elssuni 3833	. . . . . . . . 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 13084	. . . . . . . . 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 3192	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ x e. K ) -> x C_ X )
14		resiima 4979	. . . . . . 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 2220	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ x e. K ) -> ( `' ( _I |` X ) " x ) = x )
17	16	eleq1d 2244	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ x e. K ) -> ( ( `' ( _I |` X ) " x ) e. J <-> x e. J ) )
18	17	ralbidva 2471	. . 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 3143	. . 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 1353   e. wcel 2146   A.wral 2453   C_ wss 3127   U.cuni 3805   _I cid 4282   `'ccnv 4619   |` cres 4622   "cima 4623   -->wf 5204   -1-1-onto->wf1o 5207   ` cfv 5208  (class class class)co 5865  TopOnctopon 13079   Cn ccn 13256

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-map 6640  df-top 13067  df-topon 13080  df-cn 13259

This theorem is referenced by:  idcn  13283