Theorem ssidcn 14682

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 14669	. . 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 5560	. . . . 5 |- ( _I |` X ) : X -1-1-onto-> X
3		f1of 5522	. . . . 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 5348	. . . . . . 7 |- `' ( _I |` X ) = ( _I |` X )
8	7	imaeq1i 5019	. . . . . 6 |- ( `' ( _I |` X ) " x ) = ( ( _I |` X ) " x )
9		elssuni 3878	. . . . . . . . 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 14487	. . . . . . . . 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 3232	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ x e. K ) -> x C_ X )
14		resiima 5040	. . . . . . 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 2250	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ x e. K ) -> ( `' ( _I |` X ) " x ) = x )
17	16	eleq1d 2274	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` X ) ) /\ x e. K ) -> ( ( `' ( _I |` X ) " x ) e. J <-> x e. J ) )
18	17	ralbidva 2502	. . 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 3182	. . 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 1373   e. wcel 2176   A.wral 2484   C_ wss 3166   U.cuni 3850   _I cid 4335   `'ccnv 4674   |` cres 4677   "cima 4678   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5944  TopOnctopon 14482   Cn ccn 14657

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-map 6737  df-top 14470  df-topon 14483  df-cn 14660

This theorem is referenced by:  idcn  14684