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Theorem ffnaov 32074
Description: An operation maps to a class to which all values belong, analogous to ffnov 6401. (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 32046 . 2  |-  ( F : ( A  X.  B ) --> C  <->  ( F  Fn  ( A  X.  B
)  /\  A. w  e.  ( A  X.  B
) ( F''' w )  e.  C ) )
2 afveq2 32010 . . . . . 6  |-  ( w  =  <. x ,  y
>.  ->  ( F''' w )  =  ( F''' <. x ,  y >. )
)
3 df-aov 31993 . . . . . 6  |- (( x F y))  =  ( F''' <.
x ,  y >.
)
42, 3syl6eqr 2526 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  ( F''' w )  = (( x F y))  )
54eleq1d 2536 . . . 4  |-  ( w  =  <. x ,  y
>.  ->  ( ( F''' w )  e.  C  <-> (( x F y))  e.  C
) )
65ralxp 5150 . . 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 1379    e. wcel 1767   A.wral 2817   <.cop 4039    X. cxp 5003    Fn wfn 5589   -->wf 5590  '''cafv 31989   ((caov 31990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-dfat 31991  df-afv 31992  df-aov 31993
This theorem is referenced by:  faovcl  32075
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