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Theorem sprmpt2d 6870
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 6205 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v R e )  =  ( V R E ) )
43breqd 4378 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ( x ( v R e ) y  <-> 
x ( V R E ) y ) )
54adantl 464 . . . 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 1195 . . . 4  |-  ( (
ph  /\  ( v  =  V  /\  e  =  E ) )  -> 
( ch  <->  ps )
)
85, 7anbi12d 708 . . 3  |-  ( (
ph  /\  ( v  =  V  /\  e  =  E ) )  -> 
( ( x ( v R e ) y  /\  ch )  <->  ( x ( V R E ) y  /\  ps ) ) )
98opabbidv 4430 . 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 457 . 2  |-  ( ph  ->  V  e.  _V )
1210simprd 461 . 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 6238 . . 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 659 . 2  |-  ( ph  ->  { <. x ,  y
>.  |  ( x
( V R E ) y  /\  ps ) }  e.  _V )
172, 9, 11, 12, 16ovmpt2d 6329 1  |-  ( ph  ->  ( V M E )  =  { <. x ,  y >.  |  ( x ( V R E ) y  /\  ps ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971   A.wal 1397    = wceq 1399    e. wcel 1826   _Vcvv 3034   class class class wbr 4367   {copab 4424  (class class class)co 6196    |-> cmpt2 6198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-iota 5460  df-fun 5498  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201
This theorem is referenced by:  crcts  24743  cycls  24744
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