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Theorem dfopab2 6017
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 2880 . . . . 5  |-  F/ x [. ( 1st `  z
)  /  x ]. [. ( 2nd `  z
)  /  y ]. ph
2119.41 1632 . . . 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 6015 . . . . . . . 8  |-  ( z  =  <. x ,  y
>.  ->  ( [. ( 1st `  z )  /  x ]. [. ( 2nd `  z )  /  y ]. ph  <->  ph ) )
43pm5.32i 445 . . . . . . 7  |-  ( ( z  =  <. x ,  y >.  /\  [. ( 1st `  z )  /  x ]. [. ( 2nd `  z )  /  y ]. ph )  <->  ( z  =  <. x ,  y
>.  /\  ph ) )
54exbii 1552 . . . . . 6  |-  ( E. y ( z  = 
<. x ,  y >.  /\  [. ( 1st `  z
)  /  x ]. [. ( 2nd `  z
)  /  y ]. ph )  <->  E. y ( z  =  <. x ,  y
>.  /\  ph ) )
6 nfcv 2240 . . . . . . . 8  |-  F/_ y
( 1st `  z
)
7 nfsbc1v 2880 . . . . . . . 8  |-  F/ y
[. ( 2nd `  z
)  /  y ]. ph
86, 7nfsbc 2882 . . . . . . 7  |-  F/ y
[. ( 1st `  z
)  /  x ]. [. ( 2nd `  z
)  /  y ]. ph
9819.41 1632 . . . . . 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 185 . . . . 5  |-  ( E. y ( z  = 
<. x ,  y >.  /\  ph )  <->  ( E. y  z  =  <. x ,  y >.  /\  [. ( 1st `  z )  /  x ]. [. ( 2nd `  z )  /  y ]. ph ) )
1110exbii 1552 . . . 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 4539 . . . . 5  |-  ( z  e.  ( _V  X.  _V )  <->  E. x E. y 
z  =  <. x ,  y >. )
1312anbi1i 449 . . . 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 211 . . 3  |-  ( E. x E. y ( z  =  <. x ,  y >.  /\  ph ) 
<->  ( z  e.  ( _V  X.  _V )  /\  [. ( 1st `  z
)  /  x ]. [. ( 2nd `  z
)  /  y ]. ph ) )
1514abbii 2215 . 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 3930 . 2  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
17 df-rab 2384 . 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 2130 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 103    = wceq 1299   E.wex 1436    e. wcel 1448   {cab 2086   {crab 2379   _Vcvv 2641   [.wsbc 2862   <.cop 3477   {copab 3928    X. cxp 4475   ` cfv 5059   1stc1st 5967   2ndc2nd 5968
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-iota 5024  df-fun 5061  df-fv 5067  df-1st 5969  df-2nd 5970
This theorem is referenced by: (None)
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