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Theorem ovprc 6108
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 6084 . 2  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2 df-br 4213 . . . . 5  |-  ( A dom  F  B  <->  <. A ,  B >.  e.  dom  F
)
3 ovprc1.1 . . . . . 6  |-  Rel  dom  F
4 brrelex12 4915 . . . . . 6  |-  ( ( Rel  dom  F  /\  A dom  F  B )  ->  ( A  e. 
_V  /\  B  e.  _V ) )
53, 4mpan 652 . . . . 5  |-  ( A dom  F  B  -> 
( A  e.  _V  /\  B  e.  _V )
)
62, 5sylbir 205 . . . 4  |-  ( <. A ,  B >.  e. 
dom  F  ->  ( A  e.  _V  /\  B  e.  _V ) )
76con3i 129 . . 3  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  -.  <. A ,  B >.  e.  dom  F )
8 ndmfv 5755 . . 3  |-  ( -. 
<. A ,  B >.  e. 
dom  F  ->  ( F `
 <. A ,  B >. )  =  (/) )
97, 8syl 16 . 2  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( F `  <. A ,  B >. )  =  (/) )
101, 9syl5eq 2480 1  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( A F B )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   (/)c0 3628   <.cop 3817   class class class wbr 4212   dom cdm 4878   Rel wrel 4883   ` cfv 5454  (class class class)co 6081
This theorem is referenced by:  ovprc1  6109  ovprc2  6110  ovrcl  6111  bropopvvv  6426  elbasov  13513  firest  13660  psrplusg  16445  psrmulr  16448  psrvscafval  16454  mplval  16492  opsrle  16536  opsrbaslem  16538  evlval  19945  mdegfval  19985  mdetfval  27464
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-13 1727  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-pow 4377  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-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-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-dm 4888  df-iota 5418  df-fv 5462  df-ov 6084
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