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Theorem eqfnov 5638
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 5298 . 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 5209 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( F `  z )  =  ( F `  <. x ,  y >. )
)
3 fveq2 5209 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( G `  z )  =  ( G `  <. x ,  y >. )
)
42, 3eqeq12d 2096 . . . . 5  |-  ( z  =  <. x ,  y
>.  ->  ( ( F `
 z )  =  ( G `  z
)  <->  ( F `  <. x ,  y >.
)  =  ( G `
 <. x ,  y
>. ) ) )
5 df-ov 5546 . . . . . 6  |-  ( x F y )  =  ( F `  <. x ,  y >. )
6 df-ov 5546 . . . . . 6  |-  ( x G y )  =  ( G `  <. x ,  y >. )
75, 6eqeq12i 2095 . . . . 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 4507 . . 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 1285   A.wral 2349   <.cop 3409    X. cxp 4369    Fn wfn 4927   ` cfv 4932  (class class class)co 5543
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-csb 2910  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fn 4935  df-fv 4940  df-ov 5546
This theorem is referenced by:  eqfnov2  5639
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