Theorem idcn 9042

Description: A restricted identity function is a continuous function. (Contributed by FL, 31-Dec-2006.)

Hypothesis

Ref	Expression
cnpimaex.1	|- X = U.J

Assertion

Ref	Expression
idcn	|- (J e. Top -> ( _I |` X) e. (J Cn J))

Proof of Theorem idcn

Step	Hyp	Ref	Expression
1		cnpimaex.1	. . . 4 |- X = U.J
2	1, 1	iscn 9034	. . 3 |- ((J e. Top /\ J e. Top) -> (( _I |` X) e. (J Cn J) <-> (( _I |` X):X-->X /\ A.y e. J (`'( _I |` X)"y) e. J)))
3	2	anidms 480	. 2 |- (J e. Top -> (( _I |` X) e. (J Cn J) <-> (( _I |` X):X-->X /\ A.y e. J (`'( _I |` X)"y) e. J)))
4		rnresi 4281	. . . . . 6 |- ran ( _I |` X) = X
5	4	eqimssi 2668	. . . . 5 |- ran ( _I |` X) C_ X
6	5	a1i 8	. . . 4 |- (J e. Top -> ran ( _I |` X) C_ X)
7		fnresi 4529	. . . 4 |- ( _I |` X) Fn X
8	6, 7	jctil 316	. . 3 |- (J e. Top -> (( _I |` X) Fn X /\ ran ( _I |` X) C_ X))
9		df-f 4010	. . 3 |- (( _I |` X):X-->X <-> (( _I |` X) Fn X /\ ran ( _I |` X) C_ X))
10	8, 9	sylibr 217	. 2 |- (J e. Top -> ( _I |` X):X-->X)
11		funi 4452	. . . . . . . . . 10 |- Fun _I
12	11	a1i 8	. . . . . . . . 9 |- ((J e. Top /\ y e. J) -> Fun _I )
13		cnvi 4320	. . . . . . . . . . 11 |- `' _I = _I
14	13	eqcomi 1888	. . . . . . . . . 10 |- _I = `' _I
15		funeq 4441	. . . . . . . . . 10 |- ( _I = `' _I -> (Fun _I <-> Fun `' _I ))
16	14, 15	ax-mp 7	. . . . . . . . 9 |- (Fun _I <-> Fun `' _I )
17	12, 16	sylib 215	. . . . . . . 8 |- ((J e. Top /\ y e. J) -> Fun `' _I )
18		funcnvres 4487	. . . . . . . . 9 |- (Fun `' _I -> `'( _I |` X) = (`' _I |` ( _I "X)))
19		imai 4280	. . . . . . . . . . 11 |- ( _I "X) = X
20	19	a1i 8	. . . . . . . . . 10 |- (Fun `' _I -> ( _I "X) = X)
21		reseq2 4219	. . . . . . . . . 10 |- (( _I "X) = X -> (`' _I |` ( _I "X)) = (`' _I |` X))
22	20, 21	syl 12	. . . . . . . . 9 |- (Fun `' _I -> (`' _I |` ( _I "X)) = (`' _I |` X))
23	18, 22	eqtrd 1925	. . . . . . . 8 |- (Fun `' _I -> `'( _I |` X) = (`' _I |` X))
24	17, 23	syl 12	. . . . . . 7 |- ((J e. Top /\ y e. J) -> `'( _I |` X) = (`' _I |` X))
25		reseq1 4218	. . . . . . . 8 |- (`' _I = _I -> (`' _I |` X) = ( _I |` X))
26	13, 25	ax-mp 7	. . . . . . 7 |- (`' _I |` X) = ( _I |` X)
27	24, 26	syl6eq 1944	. . . . . 6 |- ((J e. Top /\ y e. J) -> `'( _I |` X) = ( _I |` X))
28	27	imaeq1d 4263	. . . . 5 |- ((J e. Top /\ y e. J) -> (`'( _I |` X)"y) = (( _I |` X)"y))
29	1	eltopss 8872	. . . . . 6 |- ((J e. Top /\ y e. J) -> y C_ X)
30		resiima 4282	. . . . . 6 |- (y C_ X -> (( _I |` X)"y) = y)
31	29, 30	syl 12	. . . . 5 |- ((J e. Top /\ y e. J) -> (( _I |` X)"y) = y)
32	28, 31	eqtrd 1925	. . . 4 |- ((J e. Top /\ y e. J) -> (`'( _I |` X)"y) = y)
33		simpr 350	. . . 4 |- ((J e. Top /\ y e. J) -> y e. J)
34	32, 33	eqeltrd 1971	. . 3 |- ((J e. Top /\ y e. J) -> (`'( _I |` X)"y) e. J)
35	34	r19.21aiva 2176	. 2 |- (J e. Top -> A.y e. J (`'( _I |` X)"y) e. J)
36	3, 10, 35	mpbir2and 802	1 |- (J e. Top -> ( _I |` X) e. (J Cn J))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105   C_ wss 2593  U.cuni 3177   _I cid 3582  `'ccnv 3985  ran crn 3987   |` cres 3988  "cima 3989  Fun wfun 3992   Fn wfn 3993  -->wf 3994  (class class class)co 4884  Topctop 8857   Cn ccn 9028

This theorem is referenced by:  metidcn 9178  ttcn 14913  extopgrp 14980  topgrpsubcnlem 14981  phtpycom 16050  phtpycolem3 16053  phtpycolem4 16054  reparpht 16065  pcohtpylem3 16082

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-oprab 4887  df-map 5383  df-cn 9030

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