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Theorem dfoprab4f 5850
Description: Operation class abstraction expressed without existential quantifiers. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypotheses
Ref Expression
dfoprab4f.x  |-  F/ x ph
dfoprab4f.y  |-  F/ y
ph
dfoprab4f.1  |-  ( w  =  <. x ,  y
>.  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
dfoprab4f  |-  { <. w ,  z >.  |  ( w  e.  ( A  X.  B )  /\  ph ) }  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  ps ) }
Distinct variable groups:    x, w, y, z    w, A, x, y    w, B, x, y    ps, w
Allowed substitution hints:    ph( x, y, z, w)    ps( x, y, z)    A( z)    B( z)

Proof of Theorem dfoprab4f
Dummy variables  u  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1462 . . . . 5  |-  F/ x  w  =  <. t ,  u >.
2 dfoprab4f.x . . . . . 6  |-  F/ x ph
3 nfs1v 1857 . . . . . 6  |-  F/ x [ t  /  x ] [ u  /  y ] ps
42, 3nfbi 1522 . . . . 5  |-  F/ x
( ph  <->  [ t  /  x ] [ u  /  y ] ps )
51, 4nfim 1505 . . . 4  |-  F/ x
( w  =  <. t ,  u >.  ->  ( ph 
<->  [ t  /  x ] [ u  /  y ] ps ) )
6 opeq1 3578 . . . . . 6  |-  ( x  =  t  ->  <. x ,  u >.  =  <. t ,  u >. )
76eqeq2d 2093 . . . . 5  |-  ( x  =  t  ->  (
w  =  <. x ,  u >.  <->  w  =  <. t ,  u >. )
)
8 sbequ12 1695 . . . . . 6  |-  ( x  =  t  ->  ( [ u  /  y ] ps  <->  [ t  /  x ] [ u  /  y ] ps ) )
98bibi2d 230 . . . . 5  |-  ( x  =  t  ->  (
( ph  <->  [ u  /  y ] ps )  <->  ( ph  <->  [ t  /  x ] [ u  /  y ] ps ) ) )
107, 9imbi12d 232 . . . 4  |-  ( x  =  t  ->  (
( w  =  <. x ,  u >.  ->  ( ph 
<->  [ u  /  y ] ps ) )  <->  ( w  =  <. t ,  u >.  ->  ( ph  <->  [ t  /  x ] [ u  /  y ] ps ) ) ) )
11 nfv 1462 . . . . . 6  |-  F/ y  w  =  <. x ,  u >.
12 dfoprab4f.y . . . . . . 7  |-  F/ y
ph
13 nfs1v 1857 . . . . . . 7  |-  F/ y [ u  /  y ] ps
1412, 13nfbi 1522 . . . . . 6  |-  F/ y ( ph  <->  [ u  /  y ] ps )
1511, 14nfim 1505 . . . . 5  |-  F/ y ( w  =  <. x ,  u >.  ->  ( ph 
<->  [ u  /  y ] ps ) )
16 opeq2 3579 . . . . . . 7  |-  ( y  =  u  ->  <. x ,  y >.  =  <. x ,  u >. )
1716eqeq2d 2093 . . . . . 6  |-  ( y  =  u  ->  (
w  =  <. x ,  y >.  <->  w  =  <. x ,  u >. ) )
18 sbequ12 1695 . . . . . . 7  |-  ( y  =  u  ->  ( ps 
<->  [ u  /  y ] ps ) )
1918bibi2d 230 . . . . . 6  |-  ( y  =  u  ->  (
( ph  <->  ps )  <->  ( ph  <->  [ u  /  y ] ps ) ) )
2017, 19imbi12d 232 . . . . 5  |-  ( y  =  u  ->  (
( w  =  <. x ,  y >.  ->  ( ph 
<->  ps ) )  <->  ( w  =  <. x ,  u >.  ->  ( ph  <->  [ u  /  y ] ps ) ) ) )
21 dfoprab4f.1 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  ( ph  <->  ps )
)
2215, 20, 21chvar 1681 . . . 4  |-  ( w  =  <. x ,  u >.  ->  ( ph  <->  [ u  /  y ] ps ) )
235, 10, 22chvar 1681 . . 3  |-  ( w  =  <. t ,  u >.  ->  ( ph  <->  [ t  /  x ] [ u  /  y ] ps ) )
2423dfoprab4 5849 . 2  |-  { <. w ,  z >.  |  ( w  e.  ( A  X.  B )  /\  ph ) }  =  { <. <. t ,  u >. ,  z >.  |  ( ( t  e.  A  /\  u  e.  B
)  /\  [ t  /  x ] [ u  /  y ] ps ) }
25 nfv 1462 . . 3  |-  F/ t ( ( x  e.  A  /\  y  e.  B )  /\  ps )
26 nfv 1462 . . 3  |-  F/ u
( ( x  e.  A  /\  y  e.  B )  /\  ps )
27 nfv 1462 . . . 4  |-  F/ x
( t  e.  A  /\  u  e.  B
)
2827, 3nfan 1498 . . 3  |-  F/ x
( ( t  e.  A  /\  u  e.  B )  /\  [
t  /  x ] [ u  /  y ] ps )
29 nfv 1462 . . . 4  |-  F/ y ( t  e.  A  /\  u  e.  B
)
3013nfsb 1864 . . . 4  |-  F/ y [ t  /  x ] [ u  /  y ] ps
3129, 30nfan 1498 . . 3  |-  F/ y ( ( t  e.  A  /\  u  e.  B )  /\  [
t  /  x ] [ u  /  y ] ps )
32 eleq1 2142 . . . . 5  |-  ( x  =  t  ->  (
x  e.  A  <->  t  e.  A ) )
33 eleq1 2142 . . . . 5  |-  ( y  =  u  ->  (
y  e.  B  <->  u  e.  B ) )
3432, 33bi2anan9 571 . . . 4  |-  ( ( x  =  t  /\  y  =  u )  ->  ( ( x  e.  A  /\  y  e.  B )  <->  ( t  e.  A  /\  u  e.  B ) ) )
3518, 8sylan9bbr 451 . . . 4  |-  ( ( x  =  t  /\  y  =  u )  ->  ( ps  <->  [ t  /  x ] [ u  /  y ] ps ) )
3634, 35anbi12d 457 . . 3  |-  ( ( x  =  t  /\  y  =  u )  ->  ( ( ( x  e.  A  /\  y  e.  B )  /\  ps ) 
<->  ( ( t  e.  A  /\  u  e.  B )  /\  [
t  /  x ] [ u  /  y ] ps ) ) )
3725, 26, 28, 31, 36cbvoprab12 5609 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B
)  /\  ps ) }  =  { <. <. t ,  u >. ,  z >.  |  ( ( t  e.  A  /\  u  e.  B )  /\  [
t  /  x ] [ u  /  y ] ps ) }
3824, 37eqtr4i 2105 1  |-  { <. w ,  z >.  |  ( w  e.  ( A  X.  B )  /\  ph ) }  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  ps ) }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285   F/wnf 1390    e. wcel 1434   [wsb 1686   <.cop 3409   {copab 3846    X. cxp 4369   {coprab 5544
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-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-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-fo 4938  df-fv 4940  df-oprab 5547  df-1st 5798  df-2nd 5799
This theorem is referenced by: (None)
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