Theorem fssres 5086

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 4926	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
2		fnssres 5032	. . . . 5 |- ( ( F Fn A /\ C C_ A ) -> ( F |` C ) Fn C )
3		resss 4653	. . . . . . 7 |- ( F |` C ) C_ F
4		rnss 4582	. . . . . . 7 |- ( ( F |` C ) C_ F -> ran ( F |` C ) C_ ran F )
5	3, 4	ax-mp 7	. . . . . 6 |- ran ( F |` C ) C_ ran F
6		sstr 3007	. . . . . 6 |- ( ( ran ( F |` C ) C_ ran F /\ ran F C_ B ) -> ran ( F |` C ) C_ B )
7	5, 6	mpan 414	. . . . 5 |- ( ran F C_ B -> ran ( F |` C ) C_ B )
8	2, 7	anim12i 331	. . . 4 |- ( ( ( F Fn A /\ C C_ A ) /\ ran F C_ B ) -> ( ( F |` C ) Fn C /\ ran ( F |` C ) C_ B ) )
9	8	an32s 532	. . 3 |- ( ( ( F Fn A /\ ran F C_ B ) /\ C C_ A ) -> ( ( F |` C ) Fn C /\ ran ( F |` C ) C_ B ) )
10	1, 9	sylanb 278	. 2 |- ( ( F : A --> B /\ C C_ A ) -> ( ( F |` C ) Fn C /\ ran ( F |` C ) C_ B ) )
11		df-f 4926	. 2 |- ( ( F |` C ) : C --> B <-> ( ( F |` C ) Fn C /\ ran ( F |` C ) C_ B ) )
12	10, 11	sylibr 132	1 |- ( ( F : A --> B /\ C C_ A ) -> ( F |` C ) : C --> B )

Syntax hints:   -> wi 4   /\ wa 102   C_ wss 2973   ran crn 4364   |` cres 4365   Fn wfn 4917   -->wf 4918

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-fun 4924  df-fn 4925  df-f 4926

This theorem is referenced by:  fssres2  5087  fresin  5088  f1ssres  5119  feqresmpt  5248  f2ndf  5867  fseq1p1m1  9111