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Theorem ovprc 6219
Description: The value of an operation when the one of the arguments is a proper class. Note: this theorem is dependent on our particular definitions of operation value, function value, and ordered pair. (Contributed by Mario Carneiro, 26-Apr-2015.)
Hypothesis
Ref Expression
ovprc1.1  |-  Rel  dom  F
Assertion
Ref Expression
ovprc  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( A F B )  =  (/) )

Proof of Theorem ovprc
StepHypRef Expression
1 df-ov 6195 . 2  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2 df-br 4393 . . . . 5  |-  ( A dom  F  B  <->  <. A ,  B >.  e.  dom  F
)
3 ovprc1.1 . . . . . 6  |-  Rel  dom  F
4 brrelex12 4976 . . . . . 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 ndmfv 5815 . . 3  |-  ( -. 
<. A ,  B >.  e. 
dom  F  ->  ( F `
 <. A ,  B >. )  =  (/) )
97, 8syl 16 . 2  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( F `  <. A ,  B >. )  =  (/) )
101, 9syl5eq 2504 1  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( A F B )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3070   (/)c0 3737   <.cop 3983   class class class wbr 4392   dom cdm 4940   Rel wrel 4945   ` cfv 5518  (class class class)co 6192
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-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631
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-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-xp 4946  df-rel 4947  df-dm 4950  df-iota 5481  df-fv 5526  df-ov 6195
This theorem is referenced by:  ovprc1  6220  ovprc2  6221  ovrcl  6222  bropopvvv  6755  supp0prc  6795  elbasov  14326  firest  14475  psrplusg  17560  psrmulr  17563  psrvscafval  17569  mplval  17610  opsrle  17666  opsrbaslem  17668  evlval  17719  matbas0pc  18397  submafval  18503  mdetfval  18510  madufval  18561  mdegfval  21649  clwwlknprop  30575
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