Theorem fssres 5373

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 5202	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
2		fnssres 5311	. . . . 5 |- ( ( F Fn A /\ C C_ A ) -> ( F |` C ) Fn C )
3		resss 4915	. . . . . . 7 |- ( F |` C ) C_ F
4		rnss 4841	. . . . . . 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 3155	. . . . . 6 |- ( ( ran ( F |` C ) C_ ran F /\ ran F C_ B ) -> ran ( F |` C ) C_ B )
7	5, 6	mpan 422	. . . . 5 |- ( ran F C_ B -> ran ( F |` C ) C_ B )
8	2, 7	anim12i 336	. . . 4 |- ( ( ( F Fn A /\ C C_ A ) /\ ran F C_ B ) -> ( ( F |` C ) Fn C /\ ran ( F |` C ) C_ B ) )
9	8	an32s 563	. . 3 |- ( ( ( F Fn A /\ ran F C_ B ) /\ C C_ A ) -> ( ( F |` C ) Fn C /\ ran ( F |` C ) C_ B ) )
10	1, 9	sylanb 282	. 2 |- ( ( F : A --> B /\ C C_ A ) -> ( ( F |` C ) Fn C /\ ran ( F |` C ) C_ B ) )
11		df-f 5202	. 2 |- ( ( F |` C ) : C --> B <-> ( ( F |` C ) Fn C /\ ran ( F |` C ) C_ B ) )
12	10, 11	sylibr 133	1 |- ( ( F : A --> B /\ C C_ A ) -> ( F |` C ) : C --> B )

Syntax hints:   -> wi 4   /\ wa 103   C_ wss 3121   ran crn 4612   |` cres 4613   Fn wfn 5193   -->wf 5194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-fun 5200  df-fn 5201  df-f 5202

This theorem is referenced by:  fssresd  5374  fssres2  5375  fresin  5376  f1ssres  5412  feqresmpt  5550  f2ndf  6205  elmapssres  6651  pmresg  6654  finomni  7116  fseq1p1m1  10050  hmeores  13109  limcdifap  13425  012of  14028  2o01f  14029