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

Theorem brovex 6853
Description: A binary relation of the value of an operation given by the "maps to" notation. (Contributed by Alexander van der Vekens, 21-Oct-2017.)
Hypotheses
Ref Expression
brovex.1  |-  O  =  ( x  e.  _V ,  y  e.  _V  |->  C )
brovex.2  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  Rel  ( V O E ) )
Assertion
Ref Expression
brovex  |-  ( F ( V O E ) P  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
) )
Distinct variable group:    x, y
Allowed substitution hints:    C( x, y)    P( x, y)    E( x, y)    F( x, y)    O( x, y)    V( x, y)

Proof of Theorem brovex
StepHypRef Expression
1 df-br 4404 . . 3  |-  ( F ( V O E ) P  <->  <. F ,  P >.  e.  ( V O E ) )
2 ne0i 3754 . . . 4  |-  ( <. F ,  P >.  e.  ( V O E )  ->  ( V O E )  =/=  (/) )
3 brovex.1 . . . . . 6  |-  O  =  ( x  e.  _V ,  y  e.  _V  |->  C )
43mpt2ndm0 6852 . . . . 5  |-  ( -.  ( V  e.  _V  /\  E  e.  _V )  ->  ( V O E )  =  (/) )
54necon1ai 2683 . . . 4  |-  ( ( V O E )  =/=  (/)  ->  ( V  e.  _V  /\  E  e. 
_V ) )
6 brovex.2 . . . . . . 7  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  Rel  ( V O E ) )
7 brrelex12 4987 . . . . . . 7  |-  ( ( Rel  ( V O E )  /\  F
( V O E ) P )  -> 
( F  e.  _V  /\  P  e.  _V )
)
86, 7sylan 471 . . . . . 6  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  F ( V O E ) P )  ->  ( F  e. 
_V  /\  P  e.  _V ) )
9 id 22 . . . . . 6  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
)  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
108, 9syldan 470 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  F ( V O E ) P )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
1110ex 434 . . . 4  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( F ( V O E ) P  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) ) )
122, 5, 113syl 20 . . 3  |-  ( <. F ,  P >.  e.  ( V O E )  ->  ( F
( V O E ) P  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
) ) )
131, 12sylbi 195 . 2  |-  ( F ( V O E ) P  ->  ( F ( V O E ) P  -> 
( ( V  e. 
_V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) ) )
1413pm2.43i 47 1  |-  ( F ( V O E ) P  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   _Vcvv 3078   (/)c0 3748   <.cop 3994   class class class wbr 4403   Rel wrel 4956  (class class class)co 6203    |-> cmpt2 6205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-xp 4957  df-rel 4958  df-dm 4961  df-iota 5492  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208
This theorem is referenced by:  brovmpt2ex  6854
  Copyright terms: Public domain W3C validator