Theorem dfimafn 3761

Description: Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.)

Assertion

Ref	Expression
dfimafn	|- ((Fun F /\ A (_ dom F) -> (F"A) = {y | E.x e. A (F` x) = y})

Distinct variable groups:   x,y,A   x,F,y

Proof of Theorem dfimafn

Step	Hyp	Ref	Expression
1		ssel 2063	. . . . . 6 |- (A (_ dom F -> (x e. A -> x e. dom F))
2		visset 1813	. . . . . . . 8 |- y e. V
3	2	funbrfvb 3755	. . . . . . 7 |- ((Fun F /\ x e. dom F) -> ((F` x) = y <-> xFy))
4	3	ex 373	. . . . . 6 |- (Fun F -> (x e. dom F -> ((F` x) = y <-> xFy)))
5	1, 4	syl9r 58	. . . . 5 |- (Fun F -> (A (_ dom F -> (x e. A -> ((F` x) = y <-> xFy))))
6	5	imp31 362	. . . 4 |- (((Fun F /\ A (_ dom F) /\ x e. A) -> ((F` x) = y <-> xFy))
7	6	rexbidva 1660	. . 3 |- ((Fun F /\ A (_ dom F) -> (E.x e. A (F` x) = y <-> E.x e. A xFy))
8	7	abbidv 1577	. 2 |- ((Fun F /\ A (_ dom F) -> {y | E.x e. A (F` x) = y} = {y | E.x e. A xFy})
9		dfima2 3405	. 2 |- (F"A) = {y | E.x e. A xFy}
10	8, 9	syl6reqr 1526	1 |- ((Fun F /\ A (_ dom F) -> (F"A) = {y | E.x e. A (F` x) = y})

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  E.wrex 1646   (_ wss 2047   class class class wbr 2619  dom cdm 3170  "cima 3173  Fun wfun 3176  ` cfv 3182

This theorem is referenced by:  dfimafn2 3762

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-rex 1650  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-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198

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