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Theorem aovprc 30243
Description: The value of an operation when the one of the arguments is a proper class, analogous to ovprc 6228. (Contributed by Alexander van der Vekens, 26-May-2017.)
Hypothesis
Ref Expression
aovprc.1  |-  Rel  dom  F
Assertion
Ref Expression
aovprc  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  -> (( A F B))  =  _V )

Proof of Theorem aovprc
StepHypRef Expression
1 df-aov 30171 . 2  |- (( A F B))  =  ( F''' <. A ,  B >. )
2 df-br 4402 . . . . 5  |-  ( A dom  F  B  <->  <. A ,  B >.  e.  dom  F
)
3 aovprc.1 . . . . . 6  |-  Rel  dom  F
4 brrelex12 4985 . . . . . 6  |-  ( ( Rel  dom  F  /\  A dom  F  B )  ->  ( A  e. 
_V  /\  B  e.  _V ) )
53, 4mpan 670 . . . . 5  |-  ( A dom  F  B  -> 
( A  e.  _V  /\  B  e.  _V )
)
62, 5sylbir 213 . . . 4  |-  ( <. A ,  B >.  e. 
dom  F  ->  ( A  e.  _V  /\  B  e.  _V ) )
76con3i 135 . . 3  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  -.  <. A ,  B >.  e.  dom  F )
8 ndmafv 30195 . . 3  |-  ( -. 
<. A ,  B >.  e. 
dom  F  ->  ( F''' <. A ,  B >. )  =  _V )
97, 8syl 16 . 2  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( F''' <. A ,  B >. )  =  _V )
101, 9syl5eq 2507 1  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  -> (( A F B))  =  _V )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3078   <.cop 3992   class class class wbr 4401   dom cdm 4949   Rel wrel 4954  '''cafv 30167   ((caov 30168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-br 4402  df-opab 4460  df-xp 4955  df-rel 4956  df-fv 5535  df-dfat 30169  df-afv 30170  df-aov 30171
This theorem is referenced by: (None)
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