Theorem fnrnov 5987

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 5533	. 2 |- ( F Fn ( A X. B ) -> ran F = { z | E. w e. ( A X. B ) z = ( F ` w ) } )
2		fveq2 5486	. . . . . 6 |- ( w = <. x , y >. -> ( F ` w ) = ( F ` <. x , y >. ) )
3		df-ov 5845	. . . . . 6 |- ( x F y ) = ( F ` <. x , y >. )
4	2, 3	eqtr4di 2217	. . . . 5 |- ( w = <. x , y >. -> ( F ` w ) = ( x F y ) )
5	4	eqeq2d 2177	. . . 4 |- ( w = <. x , y >. -> ( z = ( F ` w ) <-> z = ( x F y ) ) )
6	5	rexxp 4748	. . 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 2282	. 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 2215	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 1343   {cab 2151   E.wrex 2445   <.cop 3579   X. cxp 4602   ran crn 4605   Fn wfn 5183   ` cfv 5188  (class class class)co 5842

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fn 5191  df-fv 5196  df-ov 5845

This theorem is referenced by:  ovelrn  5990