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Theorem sprmpt2d 6974
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.) (Revised by AV, 20-Jun-2019.)
Hypotheses
Ref Expression
sprmpt2d.1  |-  M  =  ( v  e.  _V ,  e  e.  _V  |->  { <. x ,  y
>.  |  ( x
( v R e ) y  /\  ch ) } )
sprmpt2d.2  |-  ( (
ph  /\  v  =  V  /\  e  =  E )  ->  ( ch  <->  ps ) )
sprmpt2d.3  |-  ( ph  ->  ( V  e.  _V  /\  E  e.  _V )
)
sprmpt2d.4  |-  ( ph  ->  A. x A. y
( x ( V R E ) y  ->  th ) )
sprmpt2d.5  |-  ( ph  ->  { <. x ,  y
>.  |  th }  e.  _V )
Assertion
Ref Expression
sprmpt2d  |-  ( ph  ->  ( V M E )  =  { <. x ,  y >.  |  ( x ( V R E ) y  /\  ps ) } )
Distinct variable groups:    e, E, v, x, y    R, e, v    e, V, v, x, y    ph, e,
v, x, y    ps, e, v
Allowed substitution hints:    ps( x, y)    ch( x, y, v, e)    th( x, y, v, e)    R( x, y)    M( x, y, v, e)

Proof of Theorem sprmpt2d
StepHypRef Expression
1 sprmpt2d.1 . . 3  |-  M  =  ( v  e.  _V ,  e  e.  _V  |->  { <. x ,  y
>.  |  ( x
( v R e ) y  /\  ch ) } )
21a1i 11 . 2  |-  ( ph  ->  M  =  ( v  e.  _V ,  e  e.  _V  |->  { <. x ,  y >.  |  ( x ( v R e ) y  /\  ch ) } ) )
3 oveq12 6310 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v R e )  =  ( V R E ) )
43breqd 4431 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ( x ( v R e ) y  <-> 
x ( V R E ) y ) )
54adantl 467 . . . 4  |-  ( (
ph  /\  ( v  =  V  /\  e  =  E ) )  -> 
( x ( v R e ) y  <-> 
x ( V R E ) y ) )
6 sprmpt2d.2 . . . . 5  |-  ( (
ph  /\  v  =  V  /\  e  =  E )  ->  ( ch  <->  ps ) )
763expb 1206 . . . 4  |-  ( (
ph  /\  ( v  =  V  /\  e  =  E ) )  -> 
( ch  <->  ps )
)
85, 7anbi12d 715 . . 3  |-  ( (
ph  /\  ( v  =  V  /\  e  =  E ) )  -> 
( ( x ( v R e ) y  /\  ch )  <->  ( x ( V R E ) y  /\  ps ) ) )
98opabbidv 4484 . 2  |-  ( (
ph  /\  ( v  =  V  /\  e  =  E ) )  ->  { <. x ,  y
>.  |  ( x
( v R e ) y  /\  ch ) }  =  { <. x ,  y >.  |  ( x ( V R E ) y  /\  ps ) } )
10 sprmpt2d.3 . . 3  |-  ( ph  ->  ( V  e.  _V  /\  E  e.  _V )
)
1110simpld 460 . 2  |-  ( ph  ->  V  e.  _V )
1210simprd 464 . 2  |-  ( ph  ->  E  e.  _V )
13 sprmpt2d.4 . . 3  |-  ( ph  ->  A. x A. y
( x ( V R E ) y  ->  th ) )
14 sprmpt2d.5 . . 3  |-  ( ph  ->  { <. x ,  y
>.  |  th }  e.  _V )
15 opabbrex 6343 . . 3  |-  ( ( A. x A. y
( x ( V R E ) y  ->  th )  /\  { <. x ,  y >.  |  th }  e.  _V )  ->  { <. x ,  y >.  |  ( x ( V R E ) y  /\  ps ) }  e.  _V )
1613, 14, 15syl2anc 665 . 2  |-  ( ph  ->  { <. x ,  y
>.  |  ( x
( V R E ) y  /\  ps ) }  e.  _V )
172, 9, 11, 12, 16ovmpt2d 6434 1  |-  ( ph  ->  ( V M E )  =  { <. x ,  y >.  |  ( x ( V R E ) y  /\  ps ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982   A.wal 1435    = wceq 1437    e. wcel 1868   _Vcvv 3081   class class class wbr 4420   {copab 4478  (class class class)co 6301    |-> cmpt2 6303
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 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pr 4656
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 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-iota 5561  df-fun 5599  df-fv 5605  df-ov 6304  df-oprab 6305  df-mpt2 6306
This theorem is referenced by:  crcts  25335  cycls  25336
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