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Theorem ovprc 5799
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 5770 . 2  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2 opprc 3721 . . . 4  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  =  (/) )
3 0ex 4050 . . . 4  |-  (/)  e.  _V
42, 3syl6eqel 2228 . . 3  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  e. 
_V )
5 df-br 3925 . . . . 5  |-  ( A dom  F  B  <->  <. A ,  B >.  e.  dom  F
)
6 ovprc1.1 . . . . . 6  |-  Rel  dom  F
7 brrelex12 4572 . . . . . 6  |-  ( ( Rel  dom  F  /\  A dom  F  B )  ->  ( A  e. 
_V  /\  B  e.  _V ) )
86, 7mpan 420 . . . . 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 621 . . 3  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  -.  <. A ,  B >.  e.  dom  F )
11 ndmfvg 5445 . . 3  |-  ( (
<. A ,  B >.  e. 
_V  /\  -.  <. A ,  B >.  e.  dom  F
)  ->  ( F `  <. A ,  B >. )  =  (/) )
124, 10, 11syl2anc 408 . 2  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( F `  <. A ,  B >. )  =  (/) )
131, 12syl5eq 2182 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 1331    e. wcel 1480   _Vcvv 2681   (/)c0 3358   <.cop 3525   class class class wbr 3924   dom cdm 4534   Rel wrel 4539   ` cfv 5118  (class class class)co 5767
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-xp 4540  df-rel 4541  df-dm 4544  df-iota 5083  df-fv 5126  df-ov 5770
This theorem is referenced by:  ovprc1  5800  ovprc2  5801
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