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Theorem dfopab2 6838
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 3351 . . . . 5  |-  F/ x [. ( 1st `  z
)  /  x ]. [. ( 2nd `  z
)  /  y ]. ph
2119.41 1920 . . . 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 6836 . . . . . . . 8  |-  ( z  =  <. x ,  y
>.  ->  ( [. ( 1st `  z )  /  x ]. [. ( 2nd `  z )  /  y ]. ph  <->  ph ) )
43pm5.32i 637 . . . . . . 7  |-  ( ( z  =  <. x ,  y >.  /\  [. ( 1st `  z )  /  x ]. [. ( 2nd `  z )  /  y ]. ph )  <->  ( z  =  <. x ,  y
>.  /\  ph ) )
54exbii 1644 . . . . . 6  |-  ( E. y ( z  = 
<. x ,  y >.  /\  [. ( 1st `  z
)  /  x ]. [. ( 2nd `  z
)  /  y ]. ph )  <->  E. y ( z  =  <. x ,  y
>.  /\  ph ) )
6 nfcv 2629 . . . . . . . 8  |-  F/_ y
( 1st `  z
)
7 nfsbc1v 3351 . . . . . . . 8  |-  F/ y
[. ( 2nd `  z
)  /  y ]. ph
86, 7nfsbc 3353 . . . . . . 7  |-  F/ y
[. ( 1st `  z
)  /  x ]. [. ( 2nd `  z
)  /  y ]. ph
9819.41 1920 . . . . . 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 251 . . . . 5  |-  ( E. y ( z  = 
<. x ,  y >.  /\  ph )  <->  ( E. y  z  =  <. x ,  y >.  /\  [. ( 1st `  z )  /  x ]. [. ( 2nd `  z )  /  y ]. ph ) )
1110exbii 1644 . . . 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 5058 . . . . 5  |-  ( z  e.  ( _V  X.  _V )  <->  E. x E. y 
z  =  <. x ,  y >. )
1312anbi1i 695 . . . 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 277 . . 3  |-  ( E. x E. y ( z  =  <. x ,  y >.  /\  ph ) 
<->  ( z  e.  ( _V  X.  _V )  /\  [. ( 1st `  z
)  /  x ]. [. ( 2nd `  z
)  /  y ]. ph ) )
1514abbii 2601 . 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 4506 . 2  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
17 df-rab 2823 . 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 2506 1  |-  { <. x ,  y >.  |  ph }  =  { z  e.  ( _V  X.  _V )  |  [. ( 1st `  z )  /  x ]. [. ( 2nd `  z
)  /  y ]. ph }
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   {cab 2452   {crab 2818   _Vcvv 3113   [.wsbc 3331   <.cop 4033   {copab 4504    X. cxp 4997   ` cfv 5588   1stc1st 6782   2ndc2nd 6783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fv 5596  df-1st 6784  df-2nd 6785
This theorem is referenced by: (None)
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