Theorem fnrnfv 3754

Description: The range of a function expressed as a collection of the function's values.

Assertion

Ref	Expression
fnrnfv	|- (F Fn A -> ran F = {y | E.x e. A y = (F` x)})

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

Proof of Theorem fnrnfv

Step	Hyp	Ref	Expression
1		fndm 3583	. . . . . . . . 9 |- (F Fn A -> dom F = A)
2	1	eleq2d 1539	. . . . . . . 8 |- (F Fn A -> (x e. dom F <-> x e. A))
3		visset 1810	. . . . . . . . 9 |- x e. V
4	3	opeldm 3310	. . . . . . . 8 |- (<.x, y>. e. F -> x e. dom F)
5	2, 4	syl5bi 208	. . . . . . 7 |- (F Fn A -> (<.x, y>. e. F -> x e. A))
6	5	pm4.71rd 638	. . . . . 6 |- (F Fn A -> (<.x, y>. e. F <-> (x e. A /\ <.x, y>. e. F)))
7		visset 1810	. . . . . . . . 9 |- y e. V
8	7	fnopfvb 3749	. . . . . . . 8 |- ((F Fn A /\ x e. A) -> ((F` x) = y <-> <.x, y>. e. F))
9		eqcom 1475	. . . . . . . 8 |- ((F` x) = y <-> y = (F` x))
10	8, 9	syl5rbbr 534	. . . . . . 7 |- ((F Fn A /\ x e. A) -> (<.x, y>. e. F <-> y = (F` x)))
11	10	pm5.32da 648	. . . . . 6 |- (F Fn A -> ((x e. A /\ <.x, y>. e. F) <-> (x e. A /\ y = (F` x))))
12	6, 11	bitrd 527	. . . . 5 |- (F Fn A -> (<.x, y>. e. F <-> (x e. A /\ y = (F` x))))
13	12	exbidv 1278	. . . 4 |- (F Fn A -> (E.x<.x, y>. e. F <-> E.x(x e. A /\ y = (F` x))))
14		df-rex 1648	. . . 4 |- (E.x e. A y = (F` x) <-> E.x(x e. A /\ y = (F` x)))
15	13, 14	syl6bbr 537	. . 3 |- (F Fn A -> (E.x<.x, y>. e. F <-> E.x e. A y = (F` x)))
16	15	abbidv 1575	. 2 |- (F Fn A -> {y | E.x<.x, y>. e. F} = {y | E.x e. A y = (F` x)})
17		dfrn3 3300	. 2 |- ran F = {y | E.x<.x, y>. e. F}
18	16, 17	syl5eq 1517	1 |- (F Fn A -> ran F = {y | E.x e. A y = (F` x)})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 955   e. wcel 957  E.wex 979  {cab 1462  E.wrex 1644  <.cop 2408  dom cdm 3166  ran crn 3167   Fn wfn 3173  ` cfv 3178

This theorem is referenced by:  fvelrnb 3755  fniinfv 3761  dffo3 3814  fniunfv 3860  fnrnoprval 4031  grpinvf 8041  efghgrpilem 8669

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 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-fv 3194

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