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Theorem ffnaov 30110
Description: An operation maps to a class to which all values belong, analogous to ffnov 6199. (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 30082 . 2  |-  ( F : ( A  X.  B ) --> C  <->  ( F  Fn  ( A  X.  B
)  /\  A. w  e.  ( A  X.  B
) ( F''' w )  e.  C ) )
2 afveq2 30046 . . . . . 6  |-  ( w  =  <. x ,  y
>.  ->  ( F''' w )  =  ( F''' <. x ,  y >. )
)
3 df-aov 30027 . . . . . 6  |- (( x F y))  =  ( F''' <.
x ,  y >.
)
42, 3syl6eqr 2493 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  ( F''' w )  = (( x F y))  )
54eleq1d 2509 . . . 4  |-  ( w  =  <. x ,  y
>.  ->  ( ( F''' w )  e.  C  <-> (( x F y))  e.  C
) )
65ralxp 4986 . . 3  |-  ( A. w  e.  ( A  X.  B ) ( F''' w )  e.  C  <->  A. x  e.  A  A. y  e.  B (( x F y))  e.  C
)
76anbi2i 694 . 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 ) )
81, 7bitri 249 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 setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2720   <.cop 3888    X. cxp 4843    Fn wfn 5418   -->wf 5419  '''cafv 30023   ((caov 30024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-fv 5431  df-dfat 30025  df-afv 30026  df-aov 30027
This theorem is referenced by:  faovcl  30111
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