Theorem fnssresb 5031

Description: Restriction of a function with a subclass of its domain. (Contributed by NM, 10-Oct-2007.)

Assertion

Ref	Expression
fnssresb	|- ( F Fn A -> ( ( F |` B ) Fn B <-> B C_ A ) )

Proof of Theorem fnssresb

Step	Hyp	Ref	Expression
1		df-fn 4925	. 2 |- ( ( F |` B ) Fn B <-> ( Fun ( F |` B ) /\ dom ( F |` B ) = B ) )
2		fnfun 5016	. . . . 5 |- ( F Fn A -> Fun F )
3		funres 4961	. . . . 5 |- ( Fun F -> Fun ( F |` B ) )
4	2, 3	syl 14	. . . 4 |- ( F Fn A -> Fun ( F |` B ) )
5	4	biantrurd 299	. . 3 |- ( F Fn A -> ( dom ( F |` B ) = B <-> ( Fun ( F |` B ) /\ dom ( F |` B ) = B ) ) )
6		ssdmres 4651	. . . 4 |- ( B C_ dom F <-> dom ( F |` B ) = B )
7		fndm 5018	. . . . 5 |- ( F Fn A -> dom F = A )
8	7	sseq2d 3027	. . . 4 |- ( F Fn A -> ( B C_ dom F <-> B C_ A ) )
9	6, 8	syl5bbr 192	. . 3 |- ( F Fn A -> ( dom ( F |` B ) = B <-> B C_ A ) )
10	5, 9	bitr3d 188	. 2 |- ( F Fn A -> ( ( Fun ( F |` B ) /\ dom ( F |` B ) = B ) <-> B C_ A ) )
11	1, 10	syl5bb 190	1 |- ( F Fn A -> ( ( F |` B ) Fn B <-> B C_ A ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   C_ wss 2973   dom cdm 4363   |` cres 4365   Fun wfun 4916   Fn wfn 4917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-res 4375  df-fun 4924  df-fn 4925

This theorem is referenced by:  fnssres  5032