Theorem fnrnov 6151

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 5680	. 2 |- ( F Fn ( A X. B ) -> ran F = { z | E. w e. ( A X. B ) z = ( F ` w ) } )
2		fveq2 5627	. . . . . 6 |- ( w = <. x , y >. -> ( F ` w ) = ( F ` <. x , y >. ) )
3		df-ov 6004	. . . . . 6 |- ( x F y ) = ( F ` <. x , y >. )
4	2, 3	eqtr4di 2280	. . . . 5 |- ( w = <. x , y >. -> ( F ` w ) = ( x F y ) )
5	4	eqeq2d 2241	. . . 4 |- ( w = <. x , y >. -> ( z = ( F ` w ) <-> z = ( x F y ) ) )
6	5	rexxp 4866	. . 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 2345	. 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 2278	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 1395   {cab 2215   E.wrex 2509   <.cop 3669   X. cxp 4717   ran crn 4720   Fn wfn 5313   ` cfv 5318  (class class class)co 6001

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326  df-ov 6004

This theorem is referenced by:  ovelrn  6154