Theorem fnssresb 5340

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 5231	. 2 |- ( ( F |` B ) Fn B <-> ( Fun ( F |` B ) /\ dom ( F |` B ) = B ) )
2		fnfun 5325	. . . . 5 |- ( F Fn A -> Fun F )
3		funres 5269	. . . . 5 |- ( Fun F -> Fun ( F |` B ) )
4	2, 3	syl 14	. . . 4 |- ( F Fn A -> Fun ( F |` B ) )
5	4	biantrurd 305	. . 3 |- ( F Fn A -> ( dom ( F |` B ) = B <-> ( Fun ( F |` B ) /\ dom ( F |` B ) = B ) ) )
6		ssdmres 4941	. . . 4 |- ( B C_ dom F <-> dom ( F |` B ) = B )
7		fndm 5327	. . . . 5 |- ( F Fn A -> dom F = A )
8	7	sseq2d 3197	. . . 4 |- ( F Fn A -> ( B C_ dom F <-> B C_ A ) )
9	6, 8	bitr3id 194	. . 3 |- ( F Fn A -> ( dom ( F |` B ) = B <-> B C_ A ) )
10	5, 9	bitr3d 190	. 2 |- ( F Fn A -> ( ( Fun ( F |` B ) /\ dom ( F |` B ) = B ) <-> B C_ A ) )
11	1, 10	bitrid 192	1 |- ( F Fn A -> ( ( F |` B ) Fn B <-> B C_ A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1363   C_ wss 3141   dom cdm 4638   |` cres 4640   Fun wfun 5222   Fn wfn 5223

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-res 4650  df-fun 5230  df-fn 5231

This theorem is referenced by:  fnssres  5341