Theorem fssres2 5395

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 5393	. 2 |- ( ( ( F |` A ) : A --> B /\ C C_ A ) -> ( ( F |` A ) |` C ) : C --> B )
2		resabs1 4938	. . . 4 |- ( C C_ A -> ( ( F |` A ) |` C ) = ( F |` C ) )
3	2	feq1d 5354	. . 3 |- ( C C_ A -> ( ( ( F |` A ) |` C ) : C --> B <-> ( F |` C ) : C --> B ) )
4	3	adantl 277	. 2 |- ( ( ( F |` A ) : A --> B /\ C C_ A ) -> ( ( ( F |` A ) |` C ) : C --> B <-> ( F |` C ) : C --> B ) )
5	1, 4	mpbid 147	1 |- ( ( ( F |` A ) : A --> B /\ C C_ A ) -> ( F |` C ) : C --> B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   C_ wss 3131   |` cres 4630   -->wf 5214

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-fun 5220  df-fn 5221  df-f 5222

This theorem is referenced by:  frecsuclem  6409