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Theorem dfopab2 6393
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 3172 . . . . 5  |-  F/ x [. ( 1st `  z
)  /  x ]. [. ( 2nd `  z
)  /  y ]. ph
2119.41 1900 . . . 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 6391 . . . . . . . 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 1592 . . . . . 6  |-  ( E. y ( z  = 
<. x ,  y >.  /\  [. ( 1st `  z
)  /  x ]. [. ( 2nd `  z
)  /  y ]. ph )  <->  E. y ( z  =  <. x ,  y
>.  /\  ph ) )
6 nfcv 2571 . . . . . . . 8  |-  F/_ y
( 1st `  z
)
7 nfsbc1v 3172 . . . . . . . 8  |-  F/ y
[. ( 2nd `  z
)  /  y ]. ph
86, 7nfsbc 3174 . . . . . . 7  |-  F/ y
[. ( 1st `  z
)  /  x ]. [. ( 2nd `  z
)  /  y ]. ph
9819.41 1900 . . . . . 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 1592 . . . 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 4928 . . . . 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 2547 . 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 4259 . 2  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
17 df-rab 2706 . 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 2465 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 1550    = wceq 1652    e. wcel 1725   {cab 2421   {crab 2701   _Vcvv 2948   [.wsbc 3153   <.cop 3809   {copab 4257    X. cxp 4868   ` cfv 5446   1stc1st 6339   2ndc2nd 6340
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fv 5454  df-1st 6341  df-2nd 6342
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