Theorem fnssresb 5367

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 5258	. 2 |- ( ( F |` B ) Fn B <-> ( Fun ( F |` B ) /\ dom ( F |` B ) = B ) )
2		fnfun 5352	. . . . 5 |- ( F Fn A -> Fun F )
3		funres 5296	. . . . 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 4965	. . . 4 |- ( B C_ dom F <-> dom ( F |` B ) = B )
7		fndm 5354	. . . . 5 |- ( F Fn A -> dom F = A )
8	7	sseq2d 3210	. . . 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 1364   C_ wss 3154   dom cdm 4660   |` cres 4662   Fun wfun 5249   Fn wfn 5250

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-res 4672  df-fun 5257  df-fn 5258

This theorem is referenced by:  fnssres  5368  wrdred1hash  10960  plyreres  14942