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Theorem ovprc2 6340
Description: The value of an operation when the second argument is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.)
Hypothesis
Ref Expression
ovprc1.1  |-  Rel  dom  F
Assertion
Ref Expression
ovprc2  |-  ( -.  B  e.  _V  ->  ( A F B )  =  (/) )

Proof of Theorem ovprc2
StepHypRef Expression
1 simpr 468 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  B  e.  _V )
21con3i 142 . 2  |-  ( -.  B  e.  _V  ->  -.  ( A  e.  _V  /\  B  e.  _V )
)
3 ovprc1.1 . . 3  |-  Rel  dom  F
43ovprc 6338 . 2  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( A F B )  =  (/) )
52, 4syl 17 1  |-  ( -.  B  e.  _V  ->  ( A F B )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   _Vcvv 3031   (/)c0 3722   dom cdm 4839   Rel wrel 4844  (class class class)co 6308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-xp 4845  df-rel 4846  df-dm 4849  df-iota 5553  df-fv 5597  df-ov 6311
This theorem is referenced by:  ressbasss  15259  ress0  15261  wunress  15267  0rest  15406  firest  15409  subcmn  17555  dprdval0prc  17712  psrbas  18679  psr1val  18856  vr1val  18862  ply1ascl  18928  evl1fval  18993  zrhval  19156  dsmmval2  19376  restbas  20251  resstopn  20279  deg1fval  23108  submomnd  28547  suborng  28652
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