Theorem ssidcn 15092

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 15079	. . 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 5656	. . . . 5 |- ( _I |` X ) : X -1-1-onto-> X
3		f1of 5616	. . . . 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 5432	. . . . . . 7 |- `' ( _I |` X ) = ( _I |` X )
8	7	imaeq1i 5100	. . . . . 6 |- ( `' ( _I |` X ) " x ) = ( ( _I |` X ) " x )
9		elssuni 3944	. . . . . . . . 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 14897	. . . . . . . . 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 3279	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ x e. K ) -> x C_ X )
14		resiima 5122	. . . . . . 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 2279	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ x e. K ) -> ( `' ( _I |` X ) " x ) = x )
17	16	eleq1d 2303	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ x e. K ) -> ( ( `' ( _I |` X ) " x ) e. J <-> x e. J ) )
18	17	ralbidva 2540	. . 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 3229	. . 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 1398   e. wcel 2205   A.wral 2522   C_ wss 3213   U.cuni 3916   _I cid 4411   `'ccnv 4750   |` cres 4753   "cima 4754   -->wf 5350   -1-1-onto->wf1o 5353   ` cfv 5354  (class class class)co 6052  TopOnctopon 14892   Cn ccn 15067

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-map 6886  df-top 14880  df-topon 14893  df-cn 15070

This theorem is referenced by:  idcn  15094