Theorem fnrnfv 5461

Description: The range of a function expressed as a collection of the function's values. (Contributed by NM, 20-Oct-2005.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

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		dffn5im 5460	. . 3 |- ( F Fn A -> F = ( x e. A |-> ( F ` x ) ) )
2	1	rneqd 4763	. 2 |- ( F Fn A -> ran F = ran ( x e. A |-> ( F ` x ) ) )
3		eqid 2137	. . 3 |- ( x e. A |-> ( F ` x ) ) = ( x e. A |-> ( F ` x ) )
4	3	rnmpt 4782	. 2 |- ran ( x e. A |-> ( F ` x ) ) = { y | E. x e. A y = ( F ` x ) }
5	2, 4	syl6eq 2186	1 |- ( F Fn A -> ran F = { y | E. x e. A y = ( F ` x ) } )

Syntax hints:   -> wi 4   = wceq 1331   {cab 2123   E.wrex 2415   |-> cmpt 3984   ran crn 4535   Fn wfn 5113   ` cfv 5118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fn 5121  df-fv 5126

This theorem is referenced by:  fvelrnb  5462  fniinfv  5472  dffo3  5560  fniunfv  5656  fnrnov  5909