Theorem fssres2 5067

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 5066	. 2 |- ( ( ( F |` A ) : A --> B /\ C C_ A ) -> ( ( F |` A ) |` C ) : C --> B )
2		resabs1 4640	. . . 4 |- ( C C_ A -> ( ( F |` A ) |` C ) = ( F |` C ) )
3	2	feq1d 5034	. . 3 |- ( C C_ A -> ( ( ( F |` A ) |` C ) : C --> B <-> ( F |` C ) : C --> B ) )
4	3	adantl 262	. 2 |- ( ( ( F |` A ) : A --> B /\ C C_ A ) -> ( ( ( F |` A ) |` C ) : C --> B <-> ( F |` C ) : C --> B ) )
5	1, 4	mpbid 135	1 |- ( ( ( F |` A ) : A --> B /\ C C_ A ) -> ( F |` C ) : C --> B )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   C_ wss 2917   |` cres 4347   -->wf 4898

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-fun 4904  df-fn 4905  df-f 4906

This theorem is referenced by: (None)