Theorem fnrnov 6235

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 5749	. 2 |- ( F Fn ( A X. B ) -> ran F = { z | E. w e. ( A X. B ) z = ( F ` w ) } )
2		fveq2 5695	. . . . . 6 |- ( w = <. x , y >. -> ( F ` w ) = ( F ` <. x , y >. ) )
3		df-ov 6088	. . . . . 6 |- ( x F y ) = ( F ` <. x , y >. )
4	2, 3	eqtr4di 2289	. . . . 5 |- ( w = <. x , y >. -> ( F ` w ) = ( x F y ) )
5	4	eqeq2d 2250	. . . 4 |- ( w = <. x , y >. -> ( z = ( F ` w ) <-> z = ( x F y ) ) )
6	5	rexxp 4924	. . 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 2354	. 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 2287	1 |- ( F Fn ( A X. B ) -> ran F = { z | E. x e. A E. y e. B z = ( x F y ) } )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   {cab 2224   E.wrex 2529   <.cop 3712   X. cxp 4772   ran crn 4775   Fn wfn 5372   ` cfv 5377  (class class class)co 6085

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385  df-ov 6088

This theorem is used by:  ovelrn  6238