Theorem fnrnfv 3759

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 3587	. . . . . . . . 9 |- (F Fn A -> dom F = A)
2	1	eleq2d 1541	. . . . . . . 8 |- (F Fn A -> (x e. dom F <-> x e. A))
3		visset 1813	. . . . . . . . 9 |- x e. V
4	3	opeldm 3314	. . . . . . . 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 639	. . . . . 6 |- (F Fn A -> (<.x, y>. e. F <-> (x e. A /\ <.x, y>. e. F)))
7		visset 1813	. . . . . . . . 9 |- y e. V
8	7	fnopfvb 3754	. . . . . . . 8 |- ((F Fn A /\ x e. A) -> ((F` x) = y <-> <.x, y>. e. F))
9		eqcom 1477	. . . . . . . 8 |- ((F` x) = y <-> y = (F` x))
10	8, 9	syl5rbbr 535	. . . . . . 7 |- ((F Fn A /\ x e. A) -> (<.x, y>. e. F <-> y = (F` x)))
11	10	pm5.32da 649	. . . . . 6 |- (F Fn A -> ((x e. A /\ <.x, y>. e. F) <-> (x e. A /\ y = (F` x))))
12	6, 11	bitrd 528	. . . . 5 |- (F Fn A -> (<.x, y>. e. F <-> (x e. A /\ y = (F` x))))
13	12	exbidv 1279	. . . 4 |- (F Fn A -> (E.x<.x, y>. e. F <-> E.x(x e. A /\ y = (F` x))))
14		df-rex 1650	. . . 4 |- (E.x e. A y = (F` x) <-> E.x(x e. A /\ y = (F` x)))
15	13, 14	syl6bbr 538	. . 3 |- (F Fn A -> (E.x<.x, y>. e. F <-> E.x e. A y = (F` x)))
16	15	abbidv 1577	. 2 |- (F Fn A -> {y | E.x<.x, y>. e. F} = {y | E.x e. A y = (F` x)})
17		dfrn3 3304	. 2 |- ran F = {y | E.x<.x, y>. e. F}
18	16, 17	syl5eq 1519	1 |- (F Fn A -> ran F = {y | E.x e. A y = (F` x)})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  {cab 1463  E.wrex 1646  <.cop 2411  dom cdm 3170  ran crn 3171   Fn wfn 3177  ` cfv 3182

This theorem is referenced by:  fvelrnb 3760  fniinfv 3766  dffo3 3819  fniunfv 3865  fnrnoprval 4036  grpinvf 8079  efghgrpilem 8719

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

Copyright terms: Public domain