Theorem fnssresb 4525

Description: Restriction of a function with a subclass of its domain.

Assertion

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

Proof of Theorem fnssresb

Step	Hyp	Ref	Expression
1		fnfun 4510	. . . . 5 |- (F Fn A -> Fun F)
2		funres 4459	. . . . 5 |- (Fun F -> Fun (F |` B))
3	1, 2	syl 12	. . . 4 |- (F Fn A -> Fun (F |` B))
4	3	biantrurd 796	. . 3 |- (F Fn A -> (dom ( F |` B) = B <-> (Fun (F |` B) /\ dom ( F |` B) = B)))
5		fndm 4512	. . . . 5 |- (F Fn A -> dom F = A)
6	5	sseq2d 2645	. . . 4 |- (F Fn A -> (B C_ dom F <-> B C_ A))
7		ssdmres 4235	. . . 4 |- (B C_ dom F <-> dom ( F |` B) = B)
8	6, 7	syl5bbr 593	. . 3 |- (F Fn A -> (dom ( F |` B) = B <-> B C_ A))
9	4, 8	bitr3d 589	. 2 |- (F Fn A -> ((Fun (F |` B) /\ dom ( F |` B) = B) <-> B C_ A))
10		df-fn 4009	. 2 |- ((F |` B) Fn B <-> (Fun (F |` B) /\ dom ( F |` B) = B))
11	9, 10	syl5bb 591	1 |- (F Fn A -> ((F |` B) Fn B <-> B C_ A))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   C_ wss 2593  dom cdm 3986   |` cres 3988  Fun wfun 3992   Fn wfn 3993

This theorem is referenced by:  fnssres 4526  reeff1o 8691  prj1 14395  prl2 14514  filnet 15645

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-res 4006  df-fun 4008  df-fn 4009

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