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Theorem dfoprab4f 6405
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 1629 . . . . 5  |-  F/ x  w  =  <. t ,  u >.
2 dfoprab4f.x . . . . . 6  |-  F/ x ph
3 nfs1v 2182 . . . . . 6  |-  F/ x [ t  /  x ] [ u  /  y ] ps
42, 3nfbi 1856 . . . . 5  |-  F/ x
( ph  <->  [ t  /  x ] [ u  /  y ] ps )
51, 4nfim 1832 . . . 4  |-  F/ x
( w  =  <. t ,  u >.  ->  ( ph 
<->  [ t  /  x ] [ u  /  y ] ps ) )
6 opeq1 3984 . . . . . 6  |-  ( x  =  t  ->  <. x ,  u >.  =  <. t ,  u >. )
76eqeq2d 2447 . . . . 5  |-  ( x  =  t  ->  (
w  =  <. x ,  u >.  <->  w  =  <. t ,  u >. )
)
8 sbequ12 1944 . . . . . 6  |-  ( x  =  t  ->  ( [ u  /  y ] ps  <->  [ t  /  x ] [ u  /  y ] ps ) )
98bibi2d 310 . . . . 5  |-  ( x  =  t  ->  (
( ph  <->  [ u  /  y ] ps )  <->  ( ph  <->  [ t  /  x ] [ u  /  y ] ps ) ) )
107, 9imbi12d 312 . . . 4  |-  ( x  =  t  ->  (
( w  =  <. x ,  u >.  ->  ( ph 
<->  [ u  /  y ] ps ) )  <->  ( w  =  <. t ,  u >.  ->  ( ph  <->  [ t  /  x ] [ u  /  y ] ps ) ) ) )
11 nfv 1629 . . . . . 6  |-  F/ y  w  =  <. x ,  u >.
12 dfoprab4f.y . . . . . . 7  |-  F/ y
ph
13 nfs1v 2182 . . . . . . 7  |-  F/ y [ u  /  y ] ps
1412, 13nfbi 1856 . . . . . 6  |-  F/ y ( ph  <->  [ u  /  y ] ps )
1511, 14nfim 1832 . . . . 5  |-  F/ y ( w  =  <. x ,  u >.  ->  ( ph 
<->  [ u  /  y ] ps ) )
16 opeq2 3985 . . . . . . 7  |-  ( y  =  u  ->  <. x ,  y >.  =  <. x ,  u >. )
1716eqeq2d 2447 . . . . . 6  |-  ( y  =  u  ->  (
w  =  <. x ,  y >.  <->  w  =  <. x ,  u >. ) )
18 sbequ12 1944 . . . . . . 7  |-  ( y  =  u  ->  ( ps 
<->  [ u  /  y ] ps ) )
1918bibi2d 310 . . . . . 6  |-  ( y  =  u  ->  (
( ph  <->  ps )  <->  ( ph  <->  [ u  /  y ] ps ) ) )
2017, 19imbi12d 312 . . . . 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 1968 . . . 4  |-  ( w  =  <. x ,  u >.  ->  ( ph  <->  [ u  /  y ] ps ) )
235, 10, 22chvar 1968 . . 3  |-  ( w  =  <. t ,  u >.  ->  ( ph  <->  [ t  /  x ] [ u  /  y ] ps ) )
2423dfoprab4 6404 . 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 1629 . . 3  |-  F/ t ( ( x  e.  A  /\  y  e.  B )  /\  ps )
26 nfv 1629 . . 3  |-  F/ u
( ( x  e.  A  /\  y  e.  B )  /\  ps )
27 nfv 1629 . . . 4  |-  F/ x
( t  e.  A  /\  u  e.  B
)
2827, 3nfan 1846 . . 3  |-  F/ x
( ( t  e.  A  /\  u  e.  B )  /\  [
t  /  x ] [ u  /  y ] ps )
29 nfv 1629 . . . 4  |-  F/ y ( t  e.  A  /\  u  e.  B
)
3013nfsb 2185 . . . 4  |-  F/ y [ t  /  x ] [ u  /  y ] ps
3129, 30nfan 1846 . . 3  |-  F/ y ( ( t  e.  A  /\  u  e.  B )  /\  [
t  /  x ] [ u  /  y ] ps )
32 eleq1 2496 . . . . 5  |-  ( x  =  t  ->  (
x  e.  A  <->  t  e.  A ) )
33 eleq1 2496 . . . . 5  |-  ( y  =  u  ->  (
y  e.  B  <->  u  e.  B ) )
3432, 33bi2anan9 844 . . . 4  |-  ( ( x  =  t  /\  y  =  u )  ->  ( ( x  e.  A  /\  y  e.  B )  <->  ( t  e.  A  /\  u  e.  B ) ) )
3518, 8sylan9bbr 682 . . . 4  |-  ( ( x  =  t  /\  y  =  u )  ->  ( ps  <->  [ t  /  x ] [ u  /  y ] ps ) )
3634, 35anbi12d 692 . . 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 6146 . 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 2459 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    <-> wb 177    /\ wa 359   F/wnf 1553    = wceq 1652   [wsb 1658    e. wcel 1725   <.cop 3817   {copab 4265    X. cxp 4876   {coprab 6082
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fv 5462  df-oprab 6085  df-1st 6349  df-2nd 6350
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