Theorem cnrest 12994

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 2170	. . . . 5 |- U. K = U. K
3	1, 2	cnf 12963	. . . 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 5372	. 2 |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> ( F |` A ) : A --> U. K )
7		cnvresima 5098	. . . 4 |- ( `' ( F |` A ) " o ) = ( ( `' F " o ) i^i A )
8		cntop1 12960	. . . . . . 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 12765	. . . . . . . 8 |- ( J e. Top -> X e. J )
12		ssexg 4126	. . . . . . . . 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 12979	. . . . . 6 |- ( ( F e. ( J Cn K ) /\ o e. K ) -> ( `' F " o ) e. J )
18	17	adantlr 474	. . . . 5 |- ( ( ( F e. ( J Cn K ) /\ A C_ X ) /\ o e. K ) -> ( `' F " o ) e. J )
19		elrestr 12580	. . . . 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 1233	. . . 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 2257	. . 3 |- ( ( ( F e. ( J Cn K ) /\ A C_ X ) /\ o e. K ) -> ( `' ( F |` A ) " o ) e. ( J ↾t A ) )
22	21	ralrimiva 2543	. 2 |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> A. o e. K ( `' ( F |` A ) " o ) e. ( J ↾t A ) )
23	1	toptopon 12775	. . . . 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 12930	. . . 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 12961	. . . . 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 12775	. . . 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 12956	. . 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 939	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 1348   e. wcel 2141   A.wral 2448   _Vcvv 2730   i^i cin 3120   C_ wss 3121   U.cuni 3794   `'ccnv 4608   |` cres 4611   "cima 4612   -->wf 5192   ` cfv 5196  (class class class)co 5851   ↾_t crest 12572   Topctop 12754  TopOnctopon 12767   Cn ccn 12944

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-map 6626  df-rest 12574  df-topgen 12593  df-top 12755  df-topon 12768  df-bases 12800  df-cn 12947

This theorem is referenced by:  cnmpt1res  13055  cnmpt2res  13056  hmeores  13074