MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovprc Structured version   Unicode version

Theorem ovprc 6304
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 6280 . 2  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2 df-br 4443 . . . . 5  |-  ( A dom  F  B  <->  <. A ,  B >.  e.  dom  F
)
3 ovprc1.1 . . . . . 6  |-  Rel  dom  F
4 brrelex12 5031 . . . . . 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 5883 . . 3  |-  ( -. 
<. A ,  B >.  e. 
dom  F  ->  ( F `
 <. A ,  B >. )  =  (/) )
97, 8syl 16 . 2  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  ( F `  <. A ,  B >. )  =  (/) )
101, 9syl5eq 2515 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 1374    e. wcel 1762   _Vcvv 3108   (/)c0 3780   <.cop 4028   class class class wbr 4442   dom cdm 4994   Rel wrel 4999   ` cfv 5581  (class class class)co 6277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-xp 5000  df-rel 5001  df-dm 5004  df-iota 5544  df-fv 5589  df-ov 6280
This theorem is referenced by:  ovprc1  6305  ovprc2  6306  ovrcl  6307  bropopvvv  6855  supp0prc  6896  elbasov  14529  firest  14679  psrplusg  17800  psrmulr  17803  psrvscafval  17809  mplval  17850  opsrle  17906  opsrbaslem  17908  evlval  17959  matbas0pc  18673  submafval  18843  mdetfval  18850  madufval  18901  mdegfval  22190  clwwlknprop  24436
  Copyright terms: Public domain W3C validator