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Theorem brovmpt2ex 6968
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 6880 . . 3  |-  Rel  ( V O E )
32a1i 11 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  Rel  ( V O E ) )
41, 3brovex 6967 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 370    = wceq 1437    e. wcel 1867   _Vcvv 3078   class class class wbr 4417   {copab 4474   Rel wrel 4850  (class class class)co 6296    |-> cmpt2 6298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6798  df-2nd 6799
This theorem is referenced by:  2mwlk  25120  wlkbprop  25122  trliswlk  25140  pthistrl  25173  spthispth  25174  pthdepisspth  25175  wlkdvspth  25209  crctistrl  25227  cyclispth  25228  cycliscrct  25229  cyclnspth  25230  isclwlkg  25354  clwlkiswlk  25356  iseupa  25564  eupatrl  25567
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