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Theorem dfopab2 5941
Description: A way to define an ordered-pair class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
dfopab2  |-  { <. x ,  y >.  |  ph }  =  { z  e.  ( _V  X.  _V )  |  [. ( 1st `  z )  /  x ]. [. ( 2nd `  z
)  /  y ]. ph }
Distinct variable groups:    ph, z    x, y, z
Allowed substitution hints:    ph( x, y)

Proof of Theorem dfopab2
StepHypRef Expression
1 nfsbc1v 2856 . . . . 5  |-  F/ x [. ( 1st `  z
)  /  x ]. [. ( 2nd `  z
)  /  y ]. ph
2119.41 1621 . . . 4  |-  ( E. x ( E. y 
z  =  <. x ,  y >.  /\  [. ( 1st `  z )  /  x ]. [. ( 2nd `  z )  /  y ]. ph )  <->  ( E. x E. y  z  = 
<. x ,  y >.  /\  [. ( 1st `  z
)  /  x ]. [. ( 2nd `  z
)  /  y ]. ph ) )
3 sbcopeq1a 5939 . . . . . . . 8  |-  ( z  =  <. x ,  y
>.  ->  ( [. ( 1st `  z )  /  x ]. [. ( 2nd `  z )  /  y ]. ph  <->  ph ) )
43pm5.32i 442 . . . . . . 7  |-  ( ( z  =  <. x ,  y >.  /\  [. ( 1st `  z )  /  x ]. [. ( 2nd `  z )  /  y ]. ph )  <->  ( z  =  <. x ,  y
>.  /\  ph ) )
54exbii 1541 . . . . . 6  |-  ( E. y ( z  = 
<. x ,  y >.  /\  [. ( 1st `  z
)  /  x ]. [. ( 2nd `  z
)  /  y ]. ph )  <->  E. y ( z  =  <. x ,  y
>.  /\  ph ) )
6 nfcv 2228 . . . . . . . 8  |-  F/_ y
( 1st `  z
)
7 nfsbc1v 2856 . . . . . . . 8  |-  F/ y
[. ( 2nd `  z
)  /  y ]. ph
86, 7nfsbc 2858 . . . . . . 7  |-  F/ y
[. ( 1st `  z
)  /  x ]. [. ( 2nd `  z
)  /  y ]. ph
9819.41 1621 . . . . . 6  |-  ( E. y ( z  = 
<. x ,  y >.  /\  [. ( 1st `  z
)  /  x ]. [. ( 2nd `  z
)  /  y ]. ph )  <->  ( E. y 
z  =  <. x ,  y >.  /\  [. ( 1st `  z )  /  x ]. [. ( 2nd `  z )  /  y ]. ph ) )
105, 9bitr3i 184 . . . . 5  |-  ( E. y ( z  = 
<. x ,  y >.  /\  ph )  <->  ( E. y  z  =  <. x ,  y >.  /\  [. ( 1st `  z )  /  x ]. [. ( 2nd `  z )  /  y ]. ph ) )
1110exbii 1541 . . . 4  |-  ( E. x E. y ( z  =  <. x ,  y >.  /\  ph ) 
<->  E. x ( E. y  z  =  <. x ,  y >.  /\  [. ( 1st `  z )  /  x ]. [. ( 2nd `  z )  /  y ]. ph ) )
12 elvv 4488 . . . . 5  |-  ( z  e.  ( _V  X.  _V )  <->  E. x E. y 
z  =  <. x ,  y >. )
1312anbi1i 446 . . . 4  |-  ( ( z  e.  ( _V 
X.  _V )  /\  [. ( 1st `  z )  /  x ]. [. ( 2nd `  z )  /  y ]. ph )  <->  ( E. x E. y  z  = 
<. x ,  y >.  /\  [. ( 1st `  z
)  /  x ]. [. ( 2nd `  z
)  /  y ]. ph ) )
142, 11, 133bitr4i 210 . . 3  |-  ( E. x E. y ( z  =  <. x ,  y >.  /\  ph ) 
<->  ( z  e.  ( _V  X.  _V )  /\  [. ( 1st `  z
)  /  x ]. [. ( 2nd `  z
)  /  y ]. ph ) )
1514abbii 2203 . 2  |-  { z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  ph ) }  =  { z  |  ( z  e.  ( _V 
X.  _V )  /\  [. ( 1st `  z )  /  x ]. [. ( 2nd `  z )  /  y ]. ph ) }
16 df-opab 3892 . 2  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
17 df-rab 2368 . 2  |-  { z  e.  ( _V  X.  _V )  |  [. ( 1st `  z )  /  x ]. [. ( 2nd `  z )  /  y ]. ph }  =  {
z  |  ( z  e.  ( _V  X.  _V )  /\  [. ( 1st `  z )  /  x ]. [. ( 2nd `  z )  /  y ]. ph ) }
1815, 16, 173eqtr4i 2118 1  |-  { <. x ,  y >.  |  ph }  =  { z  e.  ( _V  X.  _V )  |  [. ( 1st `  z )  /  x ]. [. ( 2nd `  z
)  /  y ]. ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1289   E.wex 1426    e. wcel 1438   {cab 2074   {crab 2363   _Vcvv 2619   [.wsbc 2838   <.cop 3444   {copab 3890    X. cxp 4426   ` cfv 5002   1stc1st 5891   2ndc2nd 5892
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-iota 4967  df-fun 5004  df-fv 5010  df-1st 5893  df-2nd 5894
This theorem is referenced by: (None)
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