Theorem cnrest 13738

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 2177	. . . . 5 |- U. K = U. K
3	1, 2	cnf 13707	. . . 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 5393	. 2 |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> ( F |` A ) : A --> U. K )
7		cnvresima 5119	. . . 4 |- ( `' ( F |` A ) " o ) = ( ( `' F " o ) i^i A )
8		cntop1 13704	. . . . . . 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 13511	. . . . . . . 8 |- ( J e. Top -> X e. J )
12		ssexg 4143	. . . . . . . . 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 13723	. . . . . 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 12696	. . . . 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 1238	. . . 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 2264	. . 3 |- ( ( ( F e. ( J Cn K ) /\ A C_ X ) /\ o e. K ) -> ( `' ( F |` A ) " o ) e. ( J ↾t A ) )
22	21	ralrimiva 2550	. 2 |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> A. o e. K ( `' ( F |` A ) " o ) e. ( J ↾t A ) )
23	1	toptopon 13521	. . . . 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 13674	. . . 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 13705	. . . . 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 13521	. . . 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 13700	. . 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 944	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 1353   e. wcel 2148   A.wral 2455   _Vcvv 2738   i^i cin 3129   C_ wss 3130   U.cuni 3810   `'ccnv 4626   |` cres 4629   "cima 4630   -->wf 5213   ` cfv 5217  (class class class)co 5875   ↾_t crest 12688   Topctop 13500  TopOnctopon 13513   Cn ccn 13688

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-map 6650  df-rest 12690  df-topgen 12709  df-top 13501  df-topon 13514  df-bases 13546  df-cn 13691

This theorem is referenced by:  cnmpt1res  13799  cnmpt2res  13800  hmeores  13818