Theorem fnrnov 6091

Description: The range of an operation expressed as a collection of the operation's values. (Contributed by NM, 29-Oct-2006.)

Assertion

Ref	Expression
fnrnov	|- ( F Fn ( A X. B ) -> ran F = { z | E. x e. A E. y e. B z = ( x F y ) } )

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

Proof of Theorem fnrnov

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnrnfv 5624	. 2 |- ( F Fn ( A X. B ) -> ran F = { z | E. w e. ( A X. B ) z = ( F ` w ) } )
2		fveq2 5575	. . . . . 6 |- ( w = <. x , y >. -> ( F ` w ) = ( F ` <. x , y >. ) )
3		df-ov 5946	. . . . . 6 |- ( x F y ) = ( F ` <. x , y >. )
4	2, 3	eqtr4di 2255	. . . . 5 |- ( w = <. x , y >. -> ( F ` w ) = ( x F y ) )
5	4	eqeq2d 2216	. . . 4 |- ( w = <. x , y >. -> ( z = ( F ` w ) <-> z = ( x F y ) ) )
6	5	rexxp 4821	. . 3 |- ( E. w e. ( A X. B ) z = ( F ` w ) <-> E. x e. A E. y e. B z = ( x F y ) )
7	6	abbii 2320	. 2 |- { z | E. w e. ( A X. B ) z = ( F ` w ) } = { z | E. x e. A E. y e. B z = ( x F y ) }
8	1, 7	eqtrdi 2253	1 |- ( F Fn ( A X. B ) -> ran F = { z | E. x e. A E. y e. B z = ( x F y ) } )

Syntax hints:   -> wi 4   = wceq 1372   {cab 2190   E.wrex 2484   <.cop 3635   X. cxp 4672   ran crn 4675   Fn wfn 5265   ` cfv 5270  (class class class)co 5943

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-csb 3093  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-iota 5231  df-fun 5272  df-fn 5273  df-fv 5278  df-ov 5946

This theorem is referenced by:  ovelrn  6094