Theorem cnrest 12776

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 2164	. . . . 5 |- U. K = U. K
3	1, 2	cnf 12745	. . . 4 |- ( F e. ( J Cn K ) -> F : X --> U. K )
4	3	adantr 274	. . 3 |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> F : X --> U. K )
5		simpr 109	. . 3 |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> A C_ X )
6	4, 5	fssresd 5358	. 2 |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> ( F |` A ) : A --> U. K )
7		cnvresima 5087	. . . 4 |- ( `' ( F |` A ) " o ) = ( ( `' F " o ) i^i A )
8		cntop1 12742	. . . . . . 7 |- ( F e. ( J Cn K ) -> J e. Top )
9	8	adantr 274	. . . . . 6 |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> J e. Top )
10	9	adantr 274	. . . . 5 |- ( ( ( F e. ( J Cn K ) /\ A C_ X ) /\ o e. K ) -> J e. Top )
11	1	topopn 12547	. . . . . . . 8 |- ( J e. Top -> X e. J )
12		ssexg 4115	. . . . . . . . 9 |- ( ( A C_ X /\ X e. J ) -> A e. _V )
13	12	ancoms 266	. . . . . . . 8 |- ( ( X e. J /\ A C_ X ) -> A e. _V )
14	11, 13	sylan 281	. . . . . . 7 |- ( ( J e. Top /\ A C_ X ) -> A e. _V )
15	8, 14	sylan 281	. . . . . 6 |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> A e. _V )
16	15	adantr 274	. . . . 5 |- ( ( ( F e. ( J Cn K ) /\ A C_ X ) /\ o e. K ) -> A e. _V )
17		cnima 12761	. . . . . 6 |- ( ( F e. ( J Cn K ) /\ o e. K ) -> ( `' F " o ) e. J )
18	17	adantlr 469	. . . . 5 |- ( ( ( F e. ( J Cn K ) /\ A C_ X ) /\ o e. K ) -> ( `' F " o ) e. J )
19		elrestr 12500	. . . . 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 1227	. . . 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 2251	. . 3 |- ( ( ( F e. ( J Cn K ) /\ A C_ X ) /\ o e. K ) -> ( `' ( F |` A ) " o ) e. ( J ↾t A ) )
22	21	ralrimiva 2537	. 2 |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> A. o e. K ( `' ( F |` A ) " o ) e. ( J ↾t A ) )
23	1	toptopon 12557	. . . . 5 |- ( J e. Top <-> J e. (TopOn ` X ) )
24	8, 23	sylib 121	. . . 4 |- ( F e. ( J Cn K ) -> J e. (TopOn ` X ) )
25		resttopon 12712	. . . 4 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( J ↾t A ) e. (TopOn ` A ) )
26	24, 25	sylan 281	. . 3 |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> ( J ↾t A ) e. (TopOn ` A ) )
27		cntop2 12743	. . . . 5 |- ( F e. ( J Cn K ) -> K e. Top )
28	27	adantr 274	. . . 4 |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> K e. Top )
29	2	toptopon 12557	. . . 4 |- ( K e. Top <-> K e. (TopOn ` U. K ) )
30	28, 29	sylib 121	. . 3 |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> K e. (TopOn ` U. K ) )
31		iscn 12738	. . 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 409	. 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 933	1 |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> ( F |` A ) e. ( ( J ↾t A ) Cn K ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1342   e. wcel 2135   A.wral 2442   _Vcvv 2721   i^i cin 3110   C_ wss 3111   U.cuni 3783   `'ccnv 4597   |` cres 4600   "cima 4601   -->wf 5178   ` cfv 5182  (class class class)co 5836   ↾_t crest 12492   Topctop 12536  TopOnctopon 12549   Cn ccn 12726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4091  ax-sep 4094  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-id 4265  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-map 6607  df-rest 12494  df-topgen 12513  df-top 12537  df-topon 12550  df-bases 12582  df-cn 12729

This theorem is referenced by:  cnmpt1res  12837  cnmpt2res  12838  hmeores  12856