Theorem fores 3666

Description: Restriction of a function.

Assertion

Ref	Expression
fores	|- ((Fun F /\ A (_ dom F) -> (F |` A):A-onto->(F"A))

Proof of Theorem fores

Step	Hyp	Ref	Expression
1		funres 3537	. . 3 |- (Fun F -> Fun (F |` A))
2	1	anim1i 334	. 2 |- ((Fun F /\ A (_ dom F) -> (Fun (F |` A) /\ A (_ dom F))
3		df-fn 3183	. . 3 |- ((F |` A) Fn A <-> (Fun (F |` A) /\ dom ( F |` A) = A))
4		df-fo 3186	. . . 4 |- ((F |` A):A-onto->(F"A) <-> ((F |` A) Fn A /\ ran ( F |` A) = (F"A)))
5		df-ima 3181	. . . . 5 |- (F"A) = ran ( F |` A)
6	5	eqcomi 1471	. . . 4 |- ran ( F |` A) = (F"A)
7	4, 6	mpbiran2 727	. . 3 |- ((F |` A):A-onto->(F"A) <-> (F |` A) Fn A)
8		ssdmres 3365	. . . 4 |- (A (_ dom F <-> dom ( F |` A) = A)
9	8	anbi2i 479	. . 3 |- ((Fun (F |` A) /\ A (_ dom F) <-> (Fun (F |` A) /\ dom ( F |` A) = A))
10	3, 7, 9	3bitr4 183	. 2 |- ((F |` A):A-onto->(F"A) <-> (Fun (F |` A) /\ A (_ dom F))
11	2, 10	sylibr 200	1 |- ((Fun F /\ A (_ dom F) -> (F |` A):A-onto->(F"A))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   (_ wss 2037  dom cdm 3160  ran crn 3161   |` cres 3162  "cima 3163  Fun wfun 3166   Fn wfn 3167  -onto->wfo 3170

This theorem is referenced by:  f1ores 3688  f1oweALT 3891  fodomfi 4540  ghsubgi 8075

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fo 3186

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