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Theorem eqfnov2 5878
Description: Two operators with the same domain are equal iff their values at each point in the domain are equal. (Contributed by Jeff Madsen, 7-Jun-2010.)
Assertion
Ref Expression
eqfnov2  |-  ( ( F  Fn  ( A  X.  B )  /\  G  Fn  ( A  X.  B ) )  -> 
( F  =  G  <->  A. x  e.  A  A. y  e.  B  ( x F y )  =  ( x G y ) ) )
Distinct variable groups:    x, A, y   
x, B, y    x, F, y    x, G, y

Proof of Theorem eqfnov2
StepHypRef Expression
1 eqfnov 5877 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  G  Fn  ( A  X.  B ) )  -> 
( F  =  G  <-> 
( ( A  X.  B )  =  ( A  X.  B )  /\  A. x  e.  A  A. y  e.  B  ( x F y )  =  ( x G y ) ) ) )
2 simpr 109 . . 3  |-  ( ( ( A  X.  B
)  =  ( A  X.  B )  /\  A. x  e.  A  A. y  e.  B  (
x F y )  =  ( x G y ) )  ->  A. x  e.  A  A. y  e.  B  ( x F y )  =  ( x G y ) )
3 eqidd 2140 . . . 4  |-  ( A. x  e.  A  A. y  e.  B  (
x F y )  =  ( x G y )  ->  ( A  X.  B )  =  ( A  X.  B
) )
43ancri 322 . . 3  |-  ( A. x  e.  A  A. y  e.  B  (
x F y )  =  ( x G y )  ->  (
( A  X.  B
)  =  ( A  X.  B )  /\  A. x  e.  A  A. y  e.  B  (
x F y )  =  ( x G y ) ) )
52, 4impbii 125 . 2  |-  ( ( ( A  X.  B
)  =  ( A  X.  B )  /\  A. x  e.  A  A. y  e.  B  (
x F y )  =  ( x G y ) )  <->  A. x  e.  A  A. y  e.  B  ( x F y )  =  ( x G y ) )
61, 5syl6bb 195 1  |-  ( ( F  Fn  ( A  X.  B )  /\  G  Fn  ( A  X.  B ) )  -> 
( F  =  G  <->  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 103    <-> wb 104    = wceq 1331   A.wral 2416    X. cxp 4537    Fn wfn 5118  (class class class)co 5774
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 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fn 5126  df-fv 5131  df-ov 5777
This theorem is referenced by:  fnmpoovd  6112  tpossym  6173
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