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Theorem fnrnov 5790
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 5351 . 2  |-  ( F  Fn  ( A  X.  B )  ->  ran  F  =  { z  |  E. w  e.  ( A  X.  B ) z  =  ( F `
 w ) } )
2 fveq2 5305 . . . . . 6  |-  ( w  =  <. x ,  y
>.  ->  ( F `  w )  =  ( F `  <. x ,  y >. )
)
3 df-ov 5655 . . . . . 6  |-  ( x F y )  =  ( F `  <. x ,  y >. )
42, 3syl6eqr 2138 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  ( F `  w )  =  ( x F y ) )
54eqeq2d 2099 . . . 4  |-  ( w  =  <. x ,  y
>.  ->  ( z  =  ( F `  w
)  <->  z  =  ( x F y ) ) )
65rexxp 4580 . . 3  |-  ( E. w  e.  ( A  X.  B ) z  =  ( F `  w )  <->  E. x  e.  A  E. y  e.  B  z  =  ( x F y ) )
76abbii 2203 . 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 2136 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 1289   {cab 2074   E.wrex 2360   <.cop 3449    X. cxp 4436   ran crn 4439    Fn wfn 5010   ` cfv 5015  (class class class)co 5652
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fn 5018  df-fv 5023  df-ov 5655
This theorem is referenced by:  ovelrn  5793
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