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Theorem eqfnov2 6394
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 6393 . 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 461 . . 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 2444 . . . 4  |-  ( A. x  e.  A  A. y  e.  B  (
x F y )  =  ( x G y )  ->  ( A  X.  B )  =  ( A  X.  B
) )
43ancri 552 . . 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 188 . 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 261 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383   A.wral 2793    X. cxp 4987    Fn wfn 5573  (class class class)co 6281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-fv 5586  df-ov 6284
This theorem is referenced by:  fnmpt2ovd  6863  tpossym  6989  uncfcurf  15382  mamuass  18777  mamudi  18778  mamudir  18779  mamuvs1  18780  mamuvs2  18781  eqmat  18799  mamulid  18816  mamurid  18817  madutpos  19017  ressprdsds  20747  isngp3  20991  xrsdsre  21188  hhip  25966
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