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Theorem brovmpt2ex 6951
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.)
Hypothesis
Ref Expression
brovmpt2ex.1  |-  O  =  ( x  e.  _V ,  y  e.  _V  |->  { <. z ,  w >.  |  ph } )
Assertion
Ref Expression
brovmpt2ex  |-  ( F ( V O E ) P  ->  (
( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V )
) )
Distinct variable group:    x, w, y, z
Allowed substitution hints:    ph( x, y, z, w)    P( x, y, z, w)    E( x, y, z, w)    F( x, y, z, w)    O( x, y, z, w)    V( x, y, z, w)

Proof of Theorem brovmpt2ex
StepHypRef Expression
1 brovmpt2ex.1 . 2  |-  O  =  ( x  e.  _V ,  y  e.  _V  |->  { <. z ,  w >.  |  ph } )
21relmpt2opab 6865 . . 3  |-  Rel  ( V O E )
32a1i 11 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  Rel  ( V O E ) )
41, 3brovex 6950 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   _Vcvv 3113   class class class wbr 4447   {copab 4504   Rel wrel 5004  (class class class)co 6284    |-> cmpt2 6286
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 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
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 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785
This theorem is referenced by:  2mwlk  24225  wlkbprop  24227  trliswlk  24245  pthistrl  24278  spthispth  24279  pthdepisspth  24280  wlkdvspth  24314  crctistrl  24332  cyclispth  24333  cycliscrct  24334  cyclnspth  24335  isclwlkg  24459  clwlkiswlk  24461  iseupa  24669  eupatrl  24672
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