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Theorem eqfnov 5733
Description: Equality of two operations is determined by their values. (Contributed by NM, 1-Sep-2005.)
Assertion
Ref Expression
eqfnov  |-  ( ( F  Fn  ( A  X.  B )  /\  G  Fn  ( C  X.  D ) )  -> 
( F  =  G  <-> 
( ( A  X.  B )  =  ( C  X.  D )  /\  A. x  e.  A  A. y  e.  B  ( x F y )  =  ( x G y ) ) ) )
Distinct variable groups:    x, y, A   
x, B, y    x, F, y    x, G, y
Allowed substitution hints:    C( x, y)    D( x, y)

Proof of Theorem eqfnov
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 eqfnfv2 5382 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  G  Fn  ( C  X.  D ) )  -> 
( F  =  G  <-> 
( ( A  X.  B )  =  ( C  X.  D )  /\  A. z  e.  ( A  X.  B
) ( F `  z )  =  ( G `  z ) ) ) )
2 fveq2 5289 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( F `  z )  =  ( F `  <. x ,  y >. )
)
3 fveq2 5289 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( G `  z )  =  ( G `  <. x ,  y >. )
)
42, 3eqeq12d 2102 . . . . 5  |-  ( z  =  <. x ,  y
>.  ->  ( ( F `
 z )  =  ( G `  z
)  <->  ( F `  <. x ,  y >.
)  =  ( G `
 <. x ,  y
>. ) ) )
5 df-ov 5637 . . . . . 6  |-  ( x F y )  =  ( F `  <. x ,  y >. )
6 df-ov 5637 . . . . . 6  |-  ( x G y )  =  ( G `  <. x ,  y >. )
75, 6eqeq12i 2101 . . . . 5  |-  ( ( x F y )  =  ( x G y )  <->  ( F `  <. x ,  y
>. )  =  ( G `  <. x ,  y >. ) )
84, 7syl6bbr 196 . . . 4  |-  ( z  =  <. x ,  y
>.  ->  ( ( F `
 z )  =  ( G `  z
)  <->  ( x F y )  =  ( x G y ) ) )
98ralxp 4567 . . 3  |-  ( A. z  e.  ( A  X.  B ) ( F `
 z )  =  ( G `  z
)  <->  A. x  e.  A  A. y  e.  B  ( x F y )  =  ( x G y ) )
109anbi2i 445 . 2  |-  ( ( ( A  X.  B
)  =  ( C  X.  D )  /\  A. z  e.  ( A  X.  B ) ( F `  z )  =  ( G `  z ) )  <->  ( ( A  X.  B )  =  ( C  X.  D
)  /\  A. x  e.  A  A. y  e.  B  ( x F y )  =  ( x G y ) ) )
111, 10syl6bb 194 1  |-  ( ( F  Fn  ( A  X.  B )  /\  G  Fn  ( C  X.  D ) )  -> 
( F  =  G  <-> 
( ( A  X.  B )  =  ( C  X.  D )  /\  A. x  e.  A  A. y  e.  B  ( x F y )  =  ( x G y ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289   A.wral 2359   <.cop 3444    X. cxp 4426    Fn wfn 4997   ` cfv 5002  (class class class)co 5634
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 3949  ax-pow 4001  ax-pr 4027
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 2839  df-csb 2932  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fn 5005  df-fv 5010  df-ov 5637
This theorem is referenced by:  eqfnov2  5734
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