Theorem cnrest 14909

Description: Continuity of a restriction from a subspace. (Contributed by Jeff Hankins, 11-Jul-2009.) (Revised by Mario Carneiro, 21-Aug-2015.)

Hypothesis

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

Assertion

Ref	Expression
cnrest	|- ( ( F e. ( J Cn K ) /\ A C_ X ) -> ( F |` A ) e. ( ( J ↾t A ) Cn K ) )

Proof of Theorem cnrest

Dummy variable o is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnrest.1	. . . . 5 |- X = U. J
2		eqid 2229	. . . . 5 |- U. K = U. K
3	1, 2	cnf 14878	. . . 4 |- ( F e. ( J Cn K ) -> F : X --> U. K )
4	3	adantr 276	. . 3 |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> F : X --> U. K )
5		simpr 110	. . 3 |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> A C_ X )
6	4, 5	fssresd 5502	. 2 |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> ( F |` A ) : A --> U. K )
7		cnvresima 5218	. . . 4 |- ( `' ( F |` A ) " o ) = ( ( `' F " o ) i^i A )
8		cntop1 14875	. . . . . . 7 |- ( F e. ( J Cn K ) -> J e. Top )
9	8	adantr 276	. . . . . 6 |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> J e. Top )
10	9	adantr 276	. . . . 5 |- ( ( ( F e. ( J Cn K ) /\ A C_ X ) /\ o e. K ) -> J e. Top )
11	1	topopn 14682	. . . . . . . 8 |- ( J e. Top -> X e. J )
12		ssexg 4223	. . . . . . . . 9 |- ( ( A C_ X /\ X e. J ) -> A e. _V )
13	12	ancoms 268	. . . . . . . 8 |- ( ( X e. J /\ A C_ X ) -> A e. _V )
14	11, 13	sylan 283	. . . . . . 7 |- ( ( J e. Top /\ A C_ X ) -> A e. _V )
15	8, 14	sylan 283	. . . . . 6 |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> A e. _V )
16	15	adantr 276	. . . . 5 |- ( ( ( F e. ( J Cn K ) /\ A C_ X ) /\ o e. K ) -> A e. _V )
17		cnima 14894	. . . . . 6 |- ( ( F e. ( J Cn K ) /\ o e. K ) -> ( `' F " o ) e. J )
18	17	adantlr 477	. . . . 5 |- ( ( ( F e. ( J Cn K ) /\ A C_ X ) /\ o e. K ) -> ( `' F " o ) e. J )
19		elrestr 13280	. . . . 5 |- ( ( J e. Top /\ A e. _V /\ ( `' F " o ) e. J ) -> ( ( `' F " o ) i^i A ) e. ( J ↾t A ) )
20	10, 16, 18, 19	syl3anc 1271	. . . 4 |- ( ( ( F e. ( J Cn K ) /\ A C_ X ) /\ o e. K ) -> ( ( `' F " o ) i^i A ) e. ( J ↾t A ) )
21	7, 20	eqeltrid 2316	. . 3 |- ( ( ( F e. ( J Cn K ) /\ A C_ X ) /\ o e. K ) -> ( `' ( F |` A ) " o ) e. ( J ↾t A ) )
22	21	ralrimiva 2603	. 2 |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> A. o e. K ( `' ( F |` A ) " o ) e. ( J ↾t A ) )
23	1	toptopon 14692	. . . . 5 |- ( J e. Top <-> J e. (TopOn ` X ) )
24	8, 23	sylib 122	. . . 4 |- ( F e. ( J Cn K ) -> J e. (TopOn ` X ) )
25		resttopon 14845	. . . 4 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( J ↾t A ) e. (TopOn ` A ) )
26	24, 25	sylan 283	. . 3 |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> ( J ↾t A ) e. (TopOn ` A ) )
27		cntop2 14876	. . . . 5 |- ( F e. ( J Cn K ) -> K e. Top )
28	27	adantr 276	. . . 4 |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> K e. Top )
29	2	toptopon 14692	. . . 4 |- ( K e. Top <-> K e. (TopOn ` U. K ) )
30	28, 29	sylib 122	. . 3 |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> K e. (TopOn ` U. K ) )
31		iscn 14871	. . 3 |- ( ( ( J ↾t A ) e. (TopOn ` A ) /\ K e. (TopOn ` U. K ) ) -> ( ( F |` A ) e. ( ( J ↾t A ) Cn K ) <-> ( ( F |` A ) : A --> U. K /\ A. o e. K ( `' ( F |` A ) " o ) e. ( J ↾t A ) ) ) )
32	26, 30, 31	syl2anc 411	. 2 |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> ( ( F |` A ) e. ( ( J ↾t A ) Cn K ) <-> ( ( F |` A ) : A --> U. K /\ A. o e. K ( `' ( F |` A ) " o ) e. ( J ↾t A ) ) ) )
33	6, 22, 32	mpbir2and 950	1 |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> ( F |` A ) e. ( ( J ↾t A ) Cn K ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   A.wral 2508   _Vcvv 2799   i^i cin 3196   C_ wss 3197   U.cuni 3888   `'ccnv 4718   |` cres 4721   "cima 4722   -->wf 5314   ` cfv 5318  (class class class)co 6001   ↾_t crest 13272   Topctop 14671  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-coll 4199  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-reu 2515  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-rest 13274  df-topgen 13293  df-top 14672  df-topon 14685  df-bases 14717  df-cn 14862

This theorem is referenced by:  cnmpt1res  14970  cnmpt2res  14971  hmeores  14989