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Theorem eqfnov2 5951
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 5950 . 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 447 . . 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 2284 . . . 4  |-  ( A. x  e.  A  A. y  e.  B  (
x F y )  =  ( x G y )  ->  ( A  X.  B )  =  ( A  X.  B
) )
43ancri 535 . . 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 180 . 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 252 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 176    /\ wa 358    = wceq 1623   A.wral 2543    X. cxp 4687    Fn wfn 5250  (class class class)co 5858
This theorem is referenced by:  tpossym  6266  uncfcurf  14013  ressprdsds  17935  isngp3  18120  xrsdsre  18316  hhip  21756  mamulid  26870  mamurid  26871  mamuass  26872  mamudi  26873  mamudir  26874  mamuvs1  26875  mamuvs2  26876
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-ov 5861
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