Theorem cnres2 33562

Description: The restriction of a continuous function to a subset is continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)

Hypotheses

Ref	Expression
cnres2.1	|- X = U. J
cnres2.2	|- Y = U. K

Assertion

Ref	Expression
cnres2	|- ( ( ( J e. Top /\ K e. Top ) /\ ( A C_ X /\ B C_ Y ) /\ ( F e. ( J Cn K ) /\ A. x e. A ( F ` x ) e. B ) ) -> ( F |` A ) e. ( ( J ↾t A ) Cn ( K ↾t B ) ) )

Distinct variable groups:   x, J   x, K   x, F   x, X   x, Y   x, A   x, B

Proof of Theorem cnres2

Step	Hyp	Ref	Expression
1		simp3l 1089	. . 3 |- ( ( ( J e. Top /\ K e. Top ) /\ ( A C_ X /\ B C_ Y ) /\ ( F e. ( J Cn K ) /\ A. x e. A ( F ` x ) e. B ) ) -> F e. ( J Cn K ) )
2		simp2l 1087	. . 3 |- ( ( ( J e. Top /\ K e. Top ) /\ ( A C_ X /\ B C_ Y ) /\ ( F e. ( J Cn K ) /\ A. x e. A ( F ` x ) e. B ) ) -> A C_ X )
3		cnres2.1	. . . 4 |- X = U. J
4	3	cnrest 21089	. . 3 |- ( ( F e. ( J Cn K ) /\ A C_ X ) -> ( F |` A ) e. ( ( J ↾t A ) Cn K ) )
5	1, 2, 4	syl2anc 693	. 2 |- ( ( ( J e. Top /\ K e. Top ) /\ ( A C_ X /\ B C_ Y ) /\ ( F e. ( J Cn K ) /\ A. x e. A ( F ` x ) e. B ) ) -> ( F |` A ) e. ( ( J ↾t A ) Cn K ) )
6		simp1r 1086	. . . 4 |- ( ( ( J e. Top /\ K e. Top ) /\ ( A C_ X /\ B C_ Y ) /\ ( F e. ( J Cn K ) /\ A. x e. A ( F ` x ) e. B ) ) -> K e. Top )
7		cnres2.2	. . . . 5 |- Y = U. K
8	7	toptopon 20722	. . . 4 |- ( K e. Top <-> K e. (TopOn ` Y ) )
9	6, 8	sylib 208	. . 3 |- ( ( ( J e. Top /\ K e. Top ) /\ ( A C_ X /\ B C_ Y ) /\ ( F e. ( J Cn K ) /\ A. x e. A ( F ` x ) e. B ) ) -> K e. (TopOn ` Y ) )
10		df-ima 5127	. . . 4 |- ( F " A ) = ran ( F |` A )
11		simp3r 1090	. . . . 5 |- ( ( ( J e. Top /\ K e. Top ) /\ ( A C_ X /\ B C_ Y ) /\ ( F e. ( J Cn K ) /\ A. x e. A ( F ` x ) e. B ) ) -> A. x e. A ( F ` x ) e. B )
12	3, 7	cnf 21050	. . . . . . 7 |- ( F e. ( J Cn K ) -> F : X --> Y )
13		ffun 6048	. . . . . . 7 |- ( F : X --> Y -> Fun F )
14	1, 12, 13	3syl 18	. . . . . 6 |- ( ( ( J e. Top /\ K e. Top ) /\ ( A C_ X /\ B C_ Y ) /\ ( F e. ( J Cn K ) /\ A. x e. A ( F ` x ) e. B ) ) -> Fun F )
15		fdm 6051	. . . . . . . 8 |- ( F : X --> Y -> dom F = X )
16	1, 12, 15	3syl 18	. . . . . . 7 |- ( ( ( J e. Top /\ K e. Top ) /\ ( A C_ X /\ B C_ Y ) /\ ( F e. ( J Cn K ) /\ A. x e. A ( F ` x ) e. B ) ) -> dom F = X )
17	2, 16	sseqtr4d 3642	. . . . . 6 |- ( ( ( J e. Top /\ K e. Top ) /\ ( A C_ X /\ B C_ Y ) /\ ( F e. ( J Cn K ) /\ A. x e. A ( F ` x ) e. B ) ) -> A C_ dom F )
18		funimass4 6247	. . . . . 6 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ B <-> A. x e. A ( F ` x ) e. B ) )
19	14, 17, 18	syl2anc 693	. . . . 5 |- ( ( ( J e. Top /\ K e. Top ) /\ ( A C_ X /\ B C_ Y ) /\ ( F e. ( J Cn K ) /\ A. x e. A ( F ` x ) e. B ) ) -> ( ( F " A ) C_ B <-> A. x e. A ( F ` x ) e. B ) )
20	11, 19	mpbird 247	. . . 4 |- ( ( ( J e. Top /\ K e. Top ) /\ ( A C_ X /\ B C_ Y ) /\ ( F e. ( J Cn K ) /\ A. x e. A ( F ` x ) e. B ) ) -> ( F " A ) C_ B )
21	10, 20	syl5eqssr 3650	. . 3 |- ( ( ( J e. Top /\ K e. Top ) /\ ( A C_ X /\ B C_ Y ) /\ ( F e. ( J Cn K ) /\ A. x e. A ( F ` x ) e. B ) ) -> ran ( F |` A ) C_ B )
22		simp2r 1088	. . 3 |- ( ( ( J e. Top /\ K e. Top ) /\ ( A C_ X /\ B C_ Y ) /\ ( F e. ( J Cn K ) /\ A. x e. A ( F ` x ) e. B ) ) -> B C_ Y )
23		cnrest2 21090	. . 3 |- ( ( K e. (TopOn ` Y ) /\ ran ( F |` A ) C_ B /\ B C_ Y ) -> ( ( F |` A ) e. ( ( J ↾t A ) Cn K ) <-> ( F |` A ) e. ( ( J ↾t A ) Cn ( K ↾t B ) ) ) )
24	9, 21, 22, 23	syl3anc 1326	. 2 |- ( ( ( J e. Top /\ K e. Top ) /\ ( A C_ X /\ B C_ Y ) /\ ( F e. ( J Cn K ) /\ A. x e. A ( F ` x ) e. B ) ) -> ( ( F |` A ) e. ( ( J ↾t A ) Cn K ) <-> ( F |` A ) e. ( ( J ↾t A ) Cn ( K ↾t B ) ) ) )
25	5, 24	mpbid 222	1 |- ( ( ( J e. Top /\ K e. Top ) /\ ( A C_ X /\ B C_ Y ) /\ ( F e. ( J Cn K ) /\ A. x e. A ( F ` x ) e. B ) ) -> ( F |` A ) e. ( ( J ↾t A ) Cn ( K ↾t B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   U.cuni 4436   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   ↾_t crest 16081   Topctop 20698  TopOnctopon 20715   Cn ccn 21028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-cn 21031

This theorem is referenced by: (None)