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Theorem eqfnov2 6178
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 6177 . 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 449 . . 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 2438 . . . 4  |-  ( A. x  e.  A  A. y  e.  B  (
x F y )  =  ( x G y )  ->  ( A  X.  B )  =  ( A  X.  B
) )
43ancri 537 . . 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 182 . 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 254 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    <-> wb 178    /\ wa 360    = wceq 1653   A.wral 2706    X. cxp 4877    Fn wfn 5450  (class class class)co 6082
This theorem is referenced by:  tpossym  6512  uncfcurf  14337  ressprdsds  18402  isngp3  18646  xrsdsre  18842  hhip  22680  mamulid  27436  mamurid  27437  mamuass  27438  mamudi  27439  mamudir  27440  mamuvs1  27441  mamuvs2  27442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-fv 5463  df-ov 6085
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