Theorem fssres 5434

Description: Restriction of a function with a subclass of its domain. (Contributed by NM, 23-Sep-2004.)

Assertion

Ref	Expression
fssres	|- ( ( F : A --> B /\ C C_ A ) -> ( F |` C ) : C --> B )

Proof of Theorem fssres

Step	Hyp	Ref	Expression
1		df-f 5263	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
2		fnssres 5372	. . . . 5 |- ( ( F Fn A /\ C C_ A ) -> ( F |` C ) Fn C )
3		resss 4971	. . . . . . 7 |- ( F |` C ) C_ F
4		rnss 4897	. . . . . . 7 |- ( ( F |` C ) C_ F -> ran ( F |` C ) C_ ran F )
5	3, 4	ax-mp 5	. . . . . 6 |- ran ( F |` C ) C_ ran F
6		sstr 3192	. . . . . 6 |- ( ( ran ( F |` C ) C_ ran F /\ ran F C_ B ) -> ran ( F |` C ) C_ B )
7	5, 6	mpan 424	. . . . 5 |- ( ran F C_ B -> ran ( F |` C ) C_ B )
8	2, 7	anim12i 338	. . . 4 |- ( ( ( F Fn A /\ C C_ A ) /\ ran F C_ B ) -> ( ( F |` C ) Fn C /\ ran ( F |` C ) C_ B ) )
9	8	an32s 568	. . 3 |- ( ( ( F Fn A /\ ran F C_ B ) /\ C C_ A ) -> ( ( F |` C ) Fn C /\ ran ( F |` C ) C_ B ) )
10	1, 9	sylanb 284	. 2 |- ( ( F : A --> B /\ C C_ A ) -> ( ( F |` C ) Fn C /\ ran ( F |` C ) C_ B ) )
11		df-f 5263	. 2 |- ( ( F |` C ) : C --> B <-> ( ( F |` C ) Fn C /\ ran ( F |` C ) C_ B ) )
12	10, 11	sylibr 134	1 |- ( ( F : A --> B /\ C C_ A ) -> ( F |` C ) : C --> B )

Syntax hints:   -> wi 4   /\ wa 104   C_ wss 3157   ran crn 4665   |` cres 4666   Fn wfn 5254   -->wf 5255

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-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-fun 5261  df-fn 5262  df-f 5263

This theorem is referenced by:  fssresd  5435  fssres2  5436  fresin  5437  f1ssres  5473  feqresmpt  5616  f2ndf  6285  elmapssres  6733  pmresg  6736  finomni  7207  fseq1p1m1  10171  seqf1oglem2  10614  wrdred1  10979  resmhm  13129  resghm  13400  hmeores  14561  limcdifap  14908  012of  15650  2o01f  15651