Theorem cnrest 13029

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 12998	. . . 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 5374	. 2 |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> ( F |` A ) : A --> U. K )
7		cnvresima 5100	. . . 4 |- ( `' ( F |` A ) " o ) = ( ( `' F " o ) i^i A )
8		cntop1 12995	. . . . . . 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 12800	. . . . . . . 8 |- ( J e. Top -> X e. J )
12		ssexg 4128	. . . . . . . . 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 13014	. . . . . 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 12587	. . . . 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 12810	. . . . 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 12965	. . . 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 12996	. . . . 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 12810	. . . 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 12991	. . 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 3796   `'ccnv 4610   |` cres 4613   "cima 4614   -->wf 5194   ` cfv 5198  (class class class)co 5853   ↾_t crest 12579   Topctop 12789  TopOnctopon 12802   Cn ccn 12979

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 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

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 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-map 6628  df-rest 12581  df-topgen 12600  df-top 12790  df-topon 12803  df-bases 12835  df-cn 12982

This theorem is referenced by:  cnmpt1res  13090  cnmpt2res  13091  hmeores  13109