Theorem fores 3681

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 3551	. . 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 3193	. . 3 |- ((F |` A) Fn A <-> (Fun (F |` A) /\ dom ( F |` A) = A))
4		df-fo 3196	. . . 4 |- ((F |` A):A-onto->(F"A) <-> ((F |` A) Fn A /\ ran ( F |` A) = (F"A)))
5		df-ima 3191	. . . . 5 |- (F"A) = ran ( F |` A)
6	5	eqcomi 1479	. . . 4 |- ran ( F |` A) = (F"A)
7	4, 6	mpbiran2 729	. . 3 |- ((F |` A):A-onto->(F"A) <-> (F |` A) Fn A)
8		ssdmres 3381	. . . 4 |- (A (_ dom F <-> dom ( F |` A) = A)
9	8	anbi2i 480	. . 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 956   (_ wss 2047  dom cdm 3170  ran crn 3171   |` cres 3172  "cima 3173  Fun wfun 3176   Fn wfn 3177  -onto->wfo 3180

This theorem is referenced by:  f1ores 3703  f1oweALT 3906  fodomfiOLD 4566  ghsubgi 8138

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fo 3196

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