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Theorem ffnaov 28039
Description: An operation maps to a class to which all values belong, analogous to ffnov 6174. (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 28011 . 2  |-  ( F : ( A  X.  B ) --> C  <->  ( F  Fn  ( A  X.  B
)  /\  A. w  e.  ( A  X.  B
) ( F''' w )  e.  C ) )
2 afveq2 27975 . . . . . 6  |-  ( w  =  <. x ,  y
>.  ->  ( F''' w )  =  ( F''' <. x ,  y >. )
)
3 df-aov 27952 . . . . . 6  |- (( x F y))  =  ( F''' <.
x ,  y >.
)
42, 3syl6eqr 2486 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  ( F''' w )  = (( x F y))  )
54eleq1d 2502 . . . 4  |-  ( w  =  <. x ,  y
>.  ->  ( ( F''' w )  e.  C  <-> (( x F y))  e.  C
) )
65ralxp 5016 . . 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 676 . 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 241 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   <.cop 3817    X. cxp 4876    Fn wfn 5449   -->wf 5450  '''cafv 27948   ((caov 27949
This theorem is referenced by:  faovcl  28040
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-dfat 27950  df-afv 27951  df-aov 27952
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