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Theorem dfopab2 6340
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 3123 . . . . 5  |-  F/ x [. ( 1st `  z
)  /  x ]. [. ( 2nd `  z
)  /  y ]. ph
2119.41 1889 . . . 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 6338 . . . . . . . 8  |-  ( z  =  <. x ,  y
>.  ->  ( [. ( 1st `  z )  /  x ]. [. ( 2nd `  z )  /  y ]. ph  <->  ph ) )
43pm5.32i 619 . . . . . . 7  |-  ( ( z  =  <. x ,  y >.  /\  [. ( 1st `  z )  /  x ]. [. ( 2nd `  z )  /  y ]. ph )  <->  ( z  =  <. x ,  y
>.  /\  ph ) )
54exbii 1589 . . . . . 6  |-  ( E. y ( z  = 
<. x ,  y >.  /\  [. ( 1st `  z
)  /  x ]. [. ( 2nd `  z
)  /  y ]. ph )  <->  E. y ( z  =  <. x ,  y
>.  /\  ph ) )
6 nfcv 2523 . . . . . . . 8  |-  F/_ y
( 1st `  z
)
7 nfsbc1v 3123 . . . . . . . 8  |-  F/ y
[. ( 2nd `  z
)  /  y ]. ph
86, 7nfsbc 3125 . . . . . . 7  |-  F/ y
[. ( 1st `  z
)  /  x ]. [. ( 2nd `  z
)  /  y ]. ph
9819.41 1889 . . . . . 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 243 . . . . 5  |-  ( E. y ( z  = 
<. x ,  y >.  /\  ph )  <->  ( E. y  z  =  <. x ,  y >.  /\  [. ( 1st `  z )  /  x ]. [. ( 2nd `  z )  /  y ]. ph ) )
1110exbii 1589 . . . 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 4876 . . . . 5  |-  ( z  e.  ( _V  X.  _V )  <->  E. x E. y 
z  =  <. x ,  y >. )
1312anbi1i 677 . . . 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 269 . . 3  |-  ( E. x E. y ( z  =  <. x ,  y >.  /\  ph ) 
<->  ( z  e.  ( _V  X.  _V )  /\  [. ( 1st `  z
)  /  x ]. [. ( 2nd `  z
)  /  y ]. ph ) )
1514abbii 2499 . 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 4208 . 2  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
17 df-rab 2658 . 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 2417 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 359   E.wex 1547    = wceq 1649    e. wcel 1717   {cab 2373   {crab 2653   _Vcvv 2899   [.wsbc 3104   <.cop 3760   {copab 4206    X. cxp 4816   ` cfv 5394   1stc1st 6286   2ndc2nd 6287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-iota 5358  df-fun 5396  df-fv 5402  df-1st 6288  df-2nd 6289
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