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Theorem dfoprab4f 6220
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 1610 . . . . 5  |-  F/ x  w  =  <. t ,  u >.
2 dfoprab4f.x . . . . . 6  |-  F/ x ph
3 nfs1v 2078 . . . . . 6  |-  F/ x [ t  /  x ] [ u  /  y ] ps
42, 3nfbi 1801 . . . . 5  |-  F/ x
( ph  <->  [ t  /  x ] [ u  /  y ] ps )
51, 4nfim 1792 . . . 4  |-  F/ x
( w  =  <. t ,  u >.  ->  ( ph 
<->  [ t  /  x ] [ u  /  y ] ps ) )
6 opeq1 3833 . . . . . 6  |-  ( x  =  t  ->  <. x ,  u >.  =  <. t ,  u >. )
76eqeq2d 2327 . . . . 5  |-  ( x  =  t  ->  (
w  =  <. x ,  u >.  <->  w  =  <. t ,  u >. )
)
8 sbequ12 1891 . . . . . 6  |-  ( x  =  t  ->  ( [ u  /  y ] ps  <->  [ t  /  x ] [ u  /  y ] ps ) )
98bibi2d 309 . . . . 5  |-  ( x  =  t  ->  (
( ph  <->  [ u  /  y ] ps )  <->  ( ph  <->  [ t  /  x ] [ u  /  y ] ps ) ) )
107, 9imbi12d 311 . . . 4  |-  ( x  =  t  ->  (
( w  =  <. x ,  u >.  ->  ( ph 
<->  [ u  /  y ] ps ) )  <->  ( w  =  <. t ,  u >.  ->  ( ph  <->  [ t  /  x ] [ u  /  y ] ps ) ) ) )
11 nfv 1610 . . . . . 6  |-  F/ y  w  =  <. x ,  u >.
12 dfoprab4f.y . . . . . . 7  |-  F/ y
ph
13 nfs1v 2078 . . . . . . 7  |-  F/ y [ u  /  y ] ps
1412, 13nfbi 1801 . . . . . 6  |-  F/ y ( ph  <->  [ u  /  y ] ps )
1511, 14nfim 1792 . . . . 5  |-  F/ y ( w  =  <. x ,  u >.  ->  ( ph 
<->  [ u  /  y ] ps ) )
16 opeq2 3834 . . . . . . 7  |-  ( y  =  u  ->  <. x ,  y >.  =  <. x ,  u >. )
1716eqeq2d 2327 . . . . . 6  |-  ( y  =  u  ->  (
w  =  <. x ,  y >.  <->  w  =  <. x ,  u >. ) )
18 sbequ12 1891 . . . . . . 7  |-  ( y  =  u  ->  ( ps 
<->  [ u  /  y ] ps ) )
1918bibi2d 309 . . . . . 6  |-  ( y  =  u  ->  (
( ph  <->  ps )  <->  ( ph  <->  [ u  /  y ] ps ) ) )
2017, 19imbi12d 311 . . . . 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 1958 . . . 4  |-  ( w  =  <. x ,  u >.  ->  ( ph  <->  [ u  /  y ] ps ) )
235, 10, 22chvar 1958 . . 3  |-  ( w  =  <. t ,  u >.  ->  ( ph  <->  [ t  /  x ] [ u  /  y ] ps ) )
2423dfoprab4 6219 . 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 1610 . . 3  |-  F/ t ( ( x  e.  A  /\  y  e.  B )  /\  ps )
26 nfv 1610 . . 3  |-  F/ u
( ( x  e.  A  /\  y  e.  B )  /\  ps )
27 nfv 1610 . . . 4  |-  F/ x
( t  e.  A  /\  u  e.  B
)
2827, 3nfan 1800 . . 3  |-  F/ x
( ( t  e.  A  /\  u  e.  B )  /\  [
t  /  x ] [ u  /  y ] ps )
29 nfv 1610 . . . 4  |-  F/ y ( t  e.  A  /\  u  e.  B
)
3013nfsb 2081 . . . 4  |-  F/ y [ t  /  x ] [ u  /  y ] ps
3129, 30nfan 1800 . . 3  |-  F/ y ( ( t  e.  A  /\  u  e.  B )  /\  [
t  /  x ] [ u  /  y ] ps )
32 eleq1 2376 . . . . 5  |-  ( x  =  t  ->  (
x  e.  A  <->  t  e.  A ) )
33 eleq1 2376 . . . . 5  |-  ( y  =  u  ->  (
y  e.  B  <->  u  e.  B ) )
3432, 33bi2anan9 843 . . . 4  |-  ( ( x  =  t  /\  y  =  u )  ->  ( ( x  e.  A  /\  y  e.  B )  <->  ( t  e.  A  /\  u  e.  B ) ) )
3518, 8sylan9bbr 681 . . . 4  |-  ( ( x  =  t  /\  y  =  u )  ->  ( ps  <->  [ t  /  x ] [ u  /  y ] ps ) )
3634, 35anbi12d 691 . . 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 5962 . 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 2339 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 176    /\ wa 358   F/wnf 1535    = wceq 1633   [wsb 1639    e. wcel 1701   <.cop 3677   {copab 4113    X. cxp 4724   {coprab 5901
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-iota 5256  df-fun 5294  df-fv 5300  df-oprab 5904  df-1st 6164  df-2nd 6165
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