Theorem fnrnfv 5625

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 5624	. . 3 |- ( F Fn A -> F = ( x e. A |-> ( F ` x ) ) )
2	1	rneqd 4907	. 2 |- ( F Fn A -> ran F = ran ( x e. A |-> ( F ` x ) ) )
3		eqid 2205	. . 3 |- ( x e. A |-> ( F ` x ) ) = ( x e. A |-> ( F ` x ) )
4	3	rnmpt 4926	. 2 |- ran ( x e. A |-> ( F ` x ) ) = { y | E. x e. A y = ( F ` x ) }
5	2, 4	eqtrdi 2254	1 |- ( F Fn A -> ran F = { y | E. x e. A y = ( F ` x ) } )

Syntax hints:   -> wi 4   = wceq 1373   {cab 2191   E.wrex 2485   |-> cmpt 4105   ran crn 4676   Fn wfn 5266   ` cfv 5271

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279

This theorem is referenced by:  fvelrnb  5626  fniinfv  5637  dffo3  5727  fniunfv  5831  fnrnov  6092