Theorem fnssresb 5407

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 5293	. 2 |- ( ( F |` B ) Fn B <-> ( Fun ( F |` B ) /\ dom ( F |` B ) = B ) )
2		fnfun 5390	. . . . 5 |- ( F Fn A -> Fun F )
3		funres 5331	. . . . 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 5000	. . . 4 |- ( B C_ dom F <-> dom ( F |` B ) = B )
7		fndm 5392	. . . . 5 |- ( F Fn A -> dom F = A )
8	7	sseq2d 3231	. . . 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 1373   C_ wss 3174   dom cdm 4693   |` cres 4695   Fun wfun 5284   Fn wfn 5285

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-res 4705  df-fun 5292  df-fn 5293

This theorem is referenced by:  fnssres  5408  wrdred1hash  11074  plyreres  15351