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Theorem eqfnov2 5608
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 5607 . 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 103 . . 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 2041 . . . 4  |-  ( A. x  e.  A  A. y  e.  B  (
x F y )  =  ( x G y )  ->  ( A  X.  B )  =  ( A  X.  B
) )
43ancri 307 . . 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 117 . 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 185 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 97    <-> wb 98    = wceq 1243   A.wral 2306    X. cxp 4343    Fn wfn 4897  (class class class)co 5512
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910  df-ov 5515
This theorem is referenced by:  tpossym  5891
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