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Theorem sprmpt2 5911
Description: The extension of a binary relation which is the value of an operation given in maps-to notation. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
Hypotheses
Ref Expression
sprmpt2.1  |-  M  =  ( v  e.  _V ,  e  e.  _V  |->  { <. f ,  p >.  |  ( f ( v W e ) p  /\  ch ) } )
sprmpt2.2  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ch  <->  ps )
)
sprmpt2.3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( f ( V W E ) p  ->  th ) )
sprmpt2.4  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  { <. f ,  p >.  |  th }  e.  _V )
Assertion
Ref Expression
sprmpt2  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V M E )  =  { <. f ,  p >.  |  ( f ( V W E ) p  /\  ps ) } )
Distinct variable groups:    e, E, f, p, v    e, V, f, p, v    e, W, v    ps, e, v
Allowed substitution hints:    ps( f, p)    ch( v, e, f, p)    th( v,
e, f, p)    M( v, e, f, p)    W( f, p)

Proof of Theorem sprmpt2
StepHypRef Expression
1 sprmpt2.1 . . 3  |-  M  =  ( v  e.  _V ,  e  e.  _V  |->  { <. f ,  p >.  |  ( f ( v W e ) p  /\  ch ) } )
21a1i 9 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  M  =  ( v  e.  _V ,  e  e.  _V  |->  { <. f ,  p >.  |  ( f ( v W e ) p  /\  ch ) } ) )
3 oveq12 5572 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v W e )  =  ( V W E ) )
43adantl 271 . . . . 5  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( v  =  V  /\  e  =  E ) )  ->  (
v W e )  =  ( V W E ) )
54breqd 3816 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( v  =  V  /\  e  =  E ) )  ->  (
f ( v W e ) p  <->  f ( V W E ) p ) )
6 sprmpt2.2 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ( ch  <->  ps )
)
76adantl 271 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( v  =  V  /\  e  =  E ) )  ->  ( ch 
<->  ps ) )
85, 7anbi12d 457 . . 3  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( v  =  V  /\  e  =  E ) )  ->  (
( f ( v W e ) p  /\  ch )  <->  ( f
( V W E ) p  /\  ps ) ) )
98opabbidv 3864 . 2  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( v  =  V  /\  e  =  E ) )  ->  { <. f ,  p >.  |  ( f ( v W e ) p  /\  ch ) }  =  { <. f ,  p >.  |  ( f ( V W E ) p  /\  ps ) } )
10 simpl 107 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  V  e.  _V )
11 simpr 108 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  E  e.  _V )
12 sprmpt2.3 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( f ( V W E ) p  ->  th ) )
13 sprmpt2.4 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  { <. f ,  p >.  |  th }  e.  _V )
1412, 13opabbrex 5600 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  { <. f ,  p >.  |  ( f ( V W E ) p  /\  ps ) }  e.  _V )
152, 9, 10, 11, 14ovmpt2d 5679 1  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V M E )  =  { <. f ,  p >.  |  ( f ( V W E ) p  /\  ps ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   _Vcvv 2610   class class class wbr 3805   {copab 3858  (class class class)co 5563    |-> cmpt2 5565
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-setind 4308
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-iota 4917  df-fun 4954  df-fv 4960  df-ov 5566  df-oprab 5567  df-mpt2 5568
This theorem is referenced by: (None)
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