Theorem fnrnfv 6242

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		dffn5 6241	. . 3 |- ( F Fn A <-> F = ( x e. A |-> ( F ` x ) ) )
2		rneq 5351	. . 3 |- ( F = ( x e. A |-> ( F ` x ) ) -> ran F = ran ( x e. A |-> ( F ` x ) ) )
3	1, 2	sylbi 207	. 2 |- ( F Fn A -> ran F = ran ( x e. A |-> ( F ` x ) ) )
4		eqid 2622	. . 3 |- ( x e. A |-> ( F ` x ) ) = ( x e. A |-> ( F ` x ) )
5	4	rnmpt 5371	. 2 |- ran ( x e. A |-> ( F ` x ) ) = { y | E. x e. A y = ( F ` x ) }
6	3, 5	syl6eq 2672	1 |- ( F Fn A -> ran F = { y | E. x e. A y = ( F ` x ) } )

Syntax hints:   -> wi 4   = wceq 1483   {cab 2608   E.wrex 2913   |-> cmpt 4729   ran crn 5115   Fn wfn 5883   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  fvelrnb  6243  fniinfv  6257  dffo3  6374  fniunfv  6505  fnrnov  6807  pwcfsdom  9405  hauscmplem  21209  grpoinvf  27386  fpwrelmapffslem  29507  meascnbl  30282  omssubadd  30362  dffo3f  39364  rnfdmpr  41300  fargshiftfo  41378