Theorem fssres2 5365

Description: Restriction of a restricted function with a subclass of its domain. (Contributed by NM, 21-Jul-2005.)

Assertion

Ref	Expression
fssres2	|- ( ( ( F |` A ) : A --> B /\ C C_ A ) -> ( F |` C ) : C --> B )

Proof of Theorem fssres2

Step	Hyp	Ref	Expression
1		fssres 5363	. 2 |- ( ( ( F |` A ) : A --> B /\ C C_ A ) -> ( ( F |` A ) |` C ) : C --> B )
2		resabs1 4913	. . . 4 |- ( C C_ A -> ( ( F |` A ) |` C ) = ( F |` C ) )
3	2	feq1d 5324	. . 3 |- ( C C_ A -> ( ( ( F |` A ) |` C ) : C --> B <-> ( F |` C ) : C --> B ) )
4	3	adantl 275	. 2 |- ( ( ( F |` A ) : A --> B /\ C C_ A ) -> ( ( ( F |` A ) |` C ) : C --> B <-> ( F |` C ) : C --> B ) )
5	1, 4	mpbid 146	1 |- ( ( ( F |` A ) : A --> B /\ C C_ A ) -> ( F |` C ) : C --> B )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   C_ wss 3116   |` cres 4606   -->wf 5184

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-fun 5190  df-fn 5191  df-f 5192

This theorem is referenced by:  frecsuclem  6374