Theorem fssres 5560

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 5376	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
2		fnssres 5491	. . . . 5 |- ( ( F Fn A /\ C C_ A ) -> ( F |` C ) Fn C )
3		resss 5082	. . . . . . 7 |- ( F |` C ) C_ F
4		rnss 5007	. . . . . . 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 3256	. . . . . 6 |- ( ( ran ( F |` C ) C_ ran F /\ ran F C_ B ) -> ran ( F |` C ) C_ B )
7	5, 6	mpan 428	. . . . 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 574	. . 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 5376	. 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 3220   ran crn 4770   |` cres 4771   Fn wfn 5367   -->wf 5368

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-fun 5374  df-fn 5375  df-f 5376

This theorem is referenced by:  fssresd  5561  fssres2  5562  fresin  5563  f1ssres  5602  feqresmpt  5751  f2ndf  6452  elmapssres  6944  pmresg  6947  mapunen  7141  finomni  7470  fseq1p1m1  10479  seqf1oglem2  10935  wrdred1  11325  resmhm  13771  resghm  14040  hmeores  15339  limcdifap  15686  012of  16937  2o01f  16938