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Theorem ffnaov 27212
Description: An operation maps to a class to which all values belong, analogous to ffnov 5990. (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 27184 . 2  |-  ( F : ( A  X.  B ) --> C  <->  ( F  Fn  ( A  X.  B
)  /\  A. w  e.  ( A  X.  B
) ( F''' w )  e.  C ) )
2 eqidd 2317 . . . . . . 7  |-  ( w  =  <. x ,  y
>.  ->  F  =  F )
3 id 19 . . . . . . 7  |-  ( w  =  <. x ,  y
>.  ->  w  =  <. x ,  y >. )
42, 3afveq12d 27146 . . . . . 6  |-  ( w  =  <. x ,  y
>.  ->  ( F''' w )  =  ( F''' <. x ,  y >. )
)
5 df-aov 27124 . . . . . 6  |- (( x F y))  =  ( F''' <.
x ,  y >.
)
64, 5syl6eqr 2366 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  ( F''' w )  = (( x F y))  )
76eleq1d 2382 . . . 4  |-  ( w  =  <. x ,  y
>.  ->  ( ( F''' w )  e.  C  <-> (( x F y))  e.  C
) )
87ralxp 4864 . . 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 1633    e. wcel 1701   A.wral 2577   <.cop 3677    X. cxp 4724    Fn wfn 5287   -->wf 5288  '''cafv 27120   ((caov 27121
This theorem is referenced by:  faovcl  27213
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-fv 5300  df-dfat 27122  df-afv 27123  df-aov 27124
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