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Theorem brovex 6962
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 4454 . . 3  |-  ( F ( V O E ) P  <->  <. F ,  P >.  e.  ( V O E ) )
2 ne0i 3796 . . . 4  |-  ( <. F ,  P >.  e.  ( V O E )  ->  ( V O E )  =/=  (/) )
3 brovex.1 . . . . . 6  |-  O  =  ( x  e.  _V ,  y  e.  _V  |->  C )
43mpt2ndm0 6511 . . . . 5  |-  ( -.  ( V  e.  _V  /\  E  e.  _V )  ->  ( V O E )  =  (/) )
54necon1ai 2698 . . . 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 5043 . . . . . . 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 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3118   (/)c0 3790   <.cop 4039   class class class wbr 4453   Rel wrel 5010  (class class class)co 6295    |-> cmpt2 6297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-xp 5011  df-rel 5012  df-dm 5015  df-iota 5557  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300
This theorem is referenced by:  brovmpt2ex  6963
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