Theorem fnssresb 6003

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 5891	. 2 |- ( ( F |` B ) Fn B <-> ( Fun ( F |` B ) /\ dom ( F |` B ) = B ) )
2		fnfun 5988	. . . . 5 |- ( F Fn A -> Fun F )
3		funres 5929	. . . . 5 |- ( Fun F -> Fun ( F |` B ) )
4	2, 3	syl 17	. . . 4 |- ( F Fn A -> Fun ( F |` B ) )
5	4	biantrurd 529	. . 3 |- ( F Fn A -> ( dom ( F |` B ) = B <-> ( Fun ( F |` B ) /\ dom ( F |` B ) = B ) ) )
6		ssdmres 5420	. . . 4 |- ( B C_ dom F <-> dom ( F |` B ) = B )
7		fndm 5990	. . . . 5 |- ( F Fn A -> dom F = A )
8	7	sseq2d 3633	. . . 4 |- ( F Fn A -> ( B C_ dom F <-> B C_ A ) )
9	6, 8	syl5bbr 274	. . 3 |- ( F Fn A -> ( dom ( F |` B ) = B <-> B C_ A ) )
10	5, 9	bitr3d 270	. 2 |- ( F Fn A -> ( ( Fun ( F |` B ) /\ dom ( F |` B ) = B ) <-> B C_ A ) )
11	1, 10	syl5bb 272	1 |- ( F Fn A -> ( ( F |` B ) Fn B <-> B C_ A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   C_ wss 3574   dom cdm 5114   |` cres 5116   Fun wfun 5882   Fn wfn 5883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-fun 5890  df-fn 5891

This theorem is referenced by:  fnssres  6004  wrdred1hash  13350  plyreres  24038  xrge0pluscn  29986  icoreresf  33200  fnbrafvb  41234  rhmsscrnghm  42026  rngcrescrhm  42085  rngcrescrhmALTV  42103