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Theorem brovmpt2ex 6760
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 6674 . . 3  |-  Rel  ( V O E )
32a1i 11 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  Rel  ( V O E ) )
41, 3brovex 6759 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 1369    e. wcel 1756   _Vcvv 2991   class class class wbr 4311   {copab 4368   Rel wrel 4864  (class class class)co 6110    e. cmpt2 6112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fv 5445  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6596  df-2nd 6597
This theorem is referenced by:  2mwlk  23446  wlkbprop  23452  trliswlk  23457  pthistrl  23490  spthispth  23491  pthdepisspth  23492  wlkdvspth  23526  crctistrl  23533  cyclispth  23534  cycliscrct  23535  cyclnspth  23536  iseupa  23605  eupatrl  23608  isclwlkg  30443  clwlkiswlk  30445
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