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Theorem ovprc 5869
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 5840 . 2  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2 opprc 3774 . . . 4  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  =  (/) )
3 0ex 4104 . . . 4  |-  (/)  e.  _V
42, 3eqeltrdi 2255 . . 3  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  e. 
_V )
5 df-br 3978 . . . . 5  |-  ( A dom  F  B  <->  <. A ,  B >.  e.  dom  F
)
6 ovprc1.1 . . . . . 6  |-  Rel  dom  F
7 brrelex12 4637 . . . . . 6  |-  ( ( Rel  dom  F  /\  A dom  F  B )  ->  ( A  e. 
_V  /\  B  e.  _V ) )
86, 7mpan 421 . . . . 5  |-  ( A dom  F  B  -> 
( A  e.  _V  /\  B  e.  _V )
)
95, 8sylbir 134 . . . 4  |-  ( <. A ,  B >.  e. 
dom  F  ->  ( A  e.  _V  /\  B  e.  _V ) )
109con3i 622 . . 3  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  -.  <. A ,  B >.  e.  dom  F )
11 ndmfvg 5512 . . 3  |-  ( (
<. A ,  B >.  e. 
_V  /\  -.  <. A ,  B >.  e.  dom  F
)  ->  ( F `  <. A ,  B >. )  =  (/) )
124, 10, 11syl2anc 409 . 2  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( F `  <. A ,  B >. )  =  (/) )
131, 12syl5eq 2209 1  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( A F B )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    = wceq 1342    e. wcel 2135   _Vcvv 2722   (/)c0 3405   <.cop 3574   class class class wbr 3977   dom cdm 4599   Rel wrel 4604   ` cfv 5183  (class class class)co 5837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-nul 4103  ax-pow 4148  ax-pr 4182
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2724  df-dif 3114  df-un 3116  df-in 3118  df-ss 3125  df-nul 3406  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-uni 3785  df-br 3978  df-opab 4039  df-xp 4605  df-rel 4606  df-dm 4609  df-iota 5148  df-fv 5191  df-ov 5840
This theorem is referenced by:  ovprc1  5870  ovprc2  5871
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