Theorem fssres2 5375

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 5373	. 2 |- ( ( ( F |` A ) : A --> B /\ C C_ A ) -> ( ( F |` A ) |` C ) : C --> B )
2		resabs1 4920	. . . 4 |- ( C C_ A -> ( ( F |` A ) |` C ) = ( F |` C ) )
3	2	feq1d 5334	. . 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 3121   |` cres 4613   -->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:  frecsuclem  6385