Theorem fssres 5421

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 5250	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
2		fnssres 5359	. . . . 5 |- ( ( F Fn A /\ C C_ A ) -> ( F |` C ) Fn C )
3		resss 4960	. . . . . . 7 |- ( F |` C ) C_ F
4		rnss 4886	. . . . . . 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 3187	. . . . . 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 5250	. 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 3153   ran crn 4656   |` cres 4657   Fn wfn 5241   -->wf 5242

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-fun 5248  df-fn 5249  df-f 5250

This theorem is referenced by:  fssresd  5422  fssres2  5423  fresin  5424  f1ssres  5460  feqresmpt  5603  f2ndf  6270  elmapssres  6718  pmresg  6721  finomni  7189  fseq1p1m1  10150  seqf1oglem2  10581  wrdred1  10946  resmhm  13049  resghm  13319  hmeores  14460  limcdifap  14793  012of  15431  2o01f  15432