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Theorem ffnaov 28059
Description: An operation maps to a class to which all values belong, analogous to ffnov 5948. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
ffnaov  |-  ( F : ( A  X.  B ) --> C  <->  ( F  Fn  ( A  X.  B
)  /\  A. x  e.  A  A. y  e.  B (( x F
y))  e.  C ) )
Distinct variable groups:    x, y, A    x, B, y    x, C, y    x, F, y

Proof of Theorem ffnaov
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 ffnafv 28033 . 2  |-  ( F : ( A  X.  B ) --> C  <->  ( F  Fn  ( A  X.  B
)  /\  A. w  e.  ( A  X.  B
) ( F''' w )  e.  C ) )
2 eqidd 2284 . . . . . . 7  |-  ( w  =  <. x ,  y
>.  ->  F  =  F )
3 id 19 . . . . . . 7  |-  ( w  =  <. x ,  y
>.  ->  w  =  <. x ,  y >. )
42, 3afveq12d 27996 . . . . . 6  |-  ( w  =  <. x ,  y
>.  ->  ( F''' w )  =  ( F''' <. x ,  y >. )
)
5 df-aov 27976 . . . . . 6  |- (( x F y))  =  ( F''' <.
x ,  y >.
)
64, 5syl6eqr 2333 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  ( F''' w )  = (( x F y))  )
76eleq1d 2349 . . . 4  |-  ( w  =  <. x ,  y
>.  ->  ( ( F''' w )  e.  C  <-> (( x F y))  e.  C
) )
87ralxp 4827 . . 3  |-  ( A. w  e.  ( A  X.  B ) ( F''' w )  e.  C  <->  A. x  e.  A  A. y  e.  B (( x F y))  e.  C
)
98anbi2i 675 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  A. w  e.  ( A  X.  B ) ( F''' w )  e.  C
)  <->  ( F  Fn  ( A  X.  B
)  /\  A. x  e.  A  A. y  e.  B (( x F
y))  e.  C ) )
101, 9bitri 240 1  |-  ( F : ( A  X.  B ) --> C  <->  ( F  Fn  ( A  X.  B
)  /\  A. x  e.  A  A. y  e.  B (( x F
y))  e.  C ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   <.cop 3643    X. cxp 4687    Fn wfn 5250   -->wf 5251  '''cafv 27972   ((caov 27973
This theorem is referenced by:  faovcl  28060
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-f 5259  df-fv 5263  df-dfat 27974  df-afv 27975  df-aov 27976
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