Theorem fnssresb 3596

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

Assertion

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

Proof of Theorem fnssresb

Step	Hyp	Ref	Expression
1		fnfun 3582	. . . . 5 |- (F Fn A -> Fun F)
2		funres 3548	. . . . 5 |- (Fun F -> Fun (F |` B))
3	1, 2	syl 10	. . . 4 |- (F Fn A -> Fun (F |` B))
4	3	biantrurd 726	. . 3 |- (F Fn A -> (dom ( F |` B) = B <-> (Fun (F |` B) /\ dom ( F |` B) = B)))
5		fndm 3584	. . . . 5 |- (F Fn A -> dom F = A)
6	5	sseq2d 2087	. . . 4 |- (F Fn A -> (B (_ dom F <-> B (_ A))
7		ssdmres 3378	. . . 4 |- (B (_ dom F <-> dom ( F |` B) = B)
8	6, 7	syl5bbr 533	. . 3 |- (F Fn A -> (dom ( F |` B) = B <-> B (_ A))
9	4, 8	bitr3d 529	. 2 |- (F Fn A -> ((Fun (F |` B) /\ dom ( F |` B) = B) <-> B (_ A))
10		df-fn 3190	. 2 |- ((F |` B) Fn B <-> (Fun (F |` B) /\ dom ( F |` B) = B))
11	9, 10	syl5bb 531	1 |- (F Fn A -> ((F |` B) Fn B <-> B (_ A))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 955   (_ wss 2045  dom cdm 3167   |` cres 3169  Fun wfun 3173   Fn wfn 3174

This theorem is referenced by:  fnssres 3597  reeff1o 7404

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-pow 2739  ax-pr 2776

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-br 2617  df-opab 2664  df-id 2832  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-res 3187  df-fun 3189  df-fn 3190

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