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Theorem dfoprab4f 6856
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 1763 . . . . 5  |-  F/ x  w  =  <. t ,  u >.
2 dfoprab4f.x . . . . . 6  |-  F/ x ph
3 nfs1v 2268 . . . . . 6  |-  F/ x [ t  /  x ] [ u  /  y ] ps
42, 3nfbi 2019 . . . . 5  |-  F/ x
( ph  <->  [ t  /  x ] [ u  /  y ] ps )
51, 4nfim 2005 . . . 4  |-  F/ x
( w  =  <. t ,  u >.  ->  ( ph 
<->  [ t  /  x ] [ u  /  y ] ps ) )
6 opeq1 4169 . . . . . 6  |-  ( x  =  t  ->  <. x ,  u >.  =  <. t ,  u >. )
76eqeq2d 2463 . . . . 5  |-  ( x  =  t  ->  (
w  =  <. x ,  u >.  <->  w  =  <. t ,  u >. )
)
8 sbequ12 2085 . . . . . 6  |-  ( x  =  t  ->  ( [ u  /  y ] ps  <->  [ t  /  x ] [ u  /  y ] ps ) )
98bibi2d 320 . . . . 5  |-  ( x  =  t  ->  (
( ph  <->  [ u  /  y ] ps )  <->  ( ph  <->  [ t  /  x ] [ u  /  y ] ps ) ) )
107, 9imbi12d 322 . . . 4  |-  ( x  =  t  ->  (
( w  =  <. x ,  u >.  ->  ( ph 
<->  [ u  /  y ] ps ) )  <->  ( w  =  <. t ,  u >.  ->  ( ph  <->  [ t  /  x ] [ u  /  y ] ps ) ) ) )
11 nfv 1763 . . . . . 6  |-  F/ y  w  =  <. x ,  u >.
12 dfoprab4f.y . . . . . . 7  |-  F/ y
ph
13 nfs1v 2268 . . . . . . 7  |-  F/ y [ u  /  y ] ps
1412, 13nfbi 2019 . . . . . 6  |-  F/ y ( ph  <->  [ u  /  y ] ps )
1511, 14nfim 2005 . . . . 5  |-  F/ y ( w  =  <. x ,  u >.  ->  ( ph 
<->  [ u  /  y ] ps ) )
16 opeq2 4170 . . . . . . 7  |-  ( y  =  u  ->  <. x ,  y >.  =  <. x ,  u >. )
1716eqeq2d 2463 . . . . . 6  |-  ( y  =  u  ->  (
w  =  <. x ,  y >.  <->  w  =  <. x ,  u >. ) )
18 sbequ12 2085 . . . . . . 7  |-  ( y  =  u  ->  ( ps 
<->  [ u  /  y ] ps ) )
1918bibi2d 320 . . . . . 6  |-  ( y  =  u  ->  (
( ph  <->  ps )  <->  ( ph  <->  [ u  /  y ] ps ) ) )
2017, 19imbi12d 322 . . . . 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 2108 . . . 4  |-  ( w  =  <. x ,  u >.  ->  ( ph  <->  [ u  /  y ] ps ) )
235, 10, 22chvar 2108 . . 3  |-  ( w  =  <. t ,  u >.  ->  ( ph  <->  [ t  /  x ] [ u  /  y ] ps ) )
2423dfoprab4 6855 . 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 1763 . . 3  |-  F/ t ( ( x  e.  A  /\  y  e.  B )  /\  ps )
26 nfv 1763 . . 3  |-  F/ u
( ( x  e.  A  /\  y  e.  B )  /\  ps )
27 nfv 1763 . . . 4  |-  F/ x
( t  e.  A  /\  u  e.  B
)
2827, 3nfan 2013 . . 3  |-  F/ x
( ( t  e.  A  /\  u  e.  B )  /\  [
t  /  x ] [ u  /  y ] ps )
29 nfv 1763 . . . 4  |-  F/ y ( t  e.  A  /\  u  e.  B
)
3013nfsb 2271 . . . 4  |-  F/ y [ t  /  x ] [ u  /  y ] ps
3129, 30nfan 2013 . . 3  |-  F/ y ( ( t  e.  A  /\  u  e.  B )  /\  [
t  /  x ] [ u  /  y ] ps )
32 eleq1 2519 . . . . 5  |-  ( x  =  t  ->  (
x  e.  A  <->  t  e.  A ) )
33 eleq1 2519 . . . . 5  |-  ( y  =  u  ->  (
y  e.  B  <->  u  e.  B ) )
3432, 33bi2anan9 885 . . . 4  |-  ( ( x  =  t  /\  y  =  u )  ->  ( ( x  e.  A  /\  y  e.  B )  <->  ( t  e.  A  /\  u  e.  B ) ) )
3518, 8sylan9bbr 708 . . . 4  |-  ( ( x  =  t  /\  y  =  u )  ->  ( ps  <->  [ t  /  x ] [ u  /  y ] ps ) )
3634, 35anbi12d 718 . . 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 6370 . 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 2478 1  |-  { <. w ,  z >.  |  ( w  e.  ( A  X.  B )  /\  ph ) }  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  ps ) }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1446   F/wnf 1669   [wsb 1799    e. wcel 1889   <.cop 3976   {copab 4463    X. cxp 4835   {coprab 6296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-iota 5549  df-fun 5587  df-fv 5593  df-oprab 6299  df-1st 6798  df-2nd 6799
This theorem is referenced by: (None)
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