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Theorem fnrnov 5909
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.
StepHypRef Expression
1 fnrnfv 5461 . 2  |-  ( F  Fn  ( A  X.  B )  ->  ran  F  =  { z  |  E. w  e.  ( A  X.  B ) z  =  ( F `
 w ) } )
2 fveq2 5414 . . . . . 6  |-  ( w  =  <. x ,  y
>.  ->  ( F `  w )  =  ( F `  <. x ,  y >. )
)
3 df-ov 5770 . . . . . 6  |-  ( x F y )  =  ( F `  <. x ,  y >. )
42, 3syl6eqr 2188 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  ( F `  w )  =  ( x F y ) )
54eqeq2d 2149 . . . 4  |-  ( w  =  <. x ,  y
>.  ->  ( z  =  ( F `  w
)  <->  z  =  ( x F y ) ) )
65rexxp 4678 . . 3  |-  ( E. w  e.  ( A  X.  B ) z  =  ( F `  w )  <->  E. x  e.  A  E. y  e.  B  z  =  ( x F y ) )
76abbii 2253 . 2  |-  { z  |  E. w  e.  ( A  X.  B
) z  =  ( F `  w ) }  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  ( x F y ) }
81, 7syl6eq 2186 1  |-  ( F  Fn  ( A  X.  B )  ->  ran  F  =  { z  |  E. x  e.  A  E. y  e.  B  z  =  ( x F y ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331   {cab 2123   E.wrex 2415   <.cop 3525    X. cxp 4532   ran crn 4535    Fn wfn 5113   ` cfv 5118  (class class class)co 5767
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fn 5121  df-fv 5126  df-ov 5770
This theorem is referenced by:  ovelrn  5912
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