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Theorem relopabVD 28950
Description: Virtual deduction proof of relopab 4993. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. relopab 4993 is relopabVD 28950 without virtual deductions and was automatically derived from relopabVD 28950.
1::  |-  (. y  =  v  ->.  y  =  v ).
2:1:  |-  (. y  =  v  ->.  <. x ,. y >.  =  <. x ,. v  >. ).
3::  |-  (. y  =  v ,. x  =  u  ->.  x  =  u ).
4:3:  |-  (. y  =  v ,. x  =  u  ->.  <. x ,. v >.  =  <.  u ,  v >. ).
5:2,4:  |-  (. y  =  v ,. x  =  u  ->.  <. x ,. y >.  =  <.  u ,  v >. ).
6:5:  |-  (. y  =  v ,. x  =  u  ->.  ( z  =  <. x ,. y  >.  ->  z  =  <. u ,  v >. ) ).
7:6:  |-  (. y  =  v  ->.  ( x  =  u  ->  ( z  =  <. x ,.  y >.  ->  z  =  <. u ,  v >. ) ) ).
8:7:  |-  ( y  =  v  ->  ( x  =  u  ->  ( z  =  <. x ,. y  >.  ->  z  =  <. u ,  v >. ) ) )
9:8:  |-  ( E. v y  =  v  ->  E. v ( x  =  u  ->  ( z  =  <. x ,  y >.  ->  z  =  <. u ,  v >. ) ) )
90::  |-  ( v  =  y  <->  y  =  v )
91:90:  |-  ( E. v v  =  y  <->  E. v y  =  v )
92::  |-  E. v v  =  y
10:91,92:  |-  E. v y  =  v
11:9,10:  |-  E. v ( x  =  u  ->  ( z  =  <. x ,. y >.  ->  z  =  <. u ,  v >. ) )
12:11:  |-  ( x  =  u  ->  E. v ( z  =  <. x ,. y >.  ->  z  =  <. u ,  v >. ) )
13::  |-  ( E. v ( z  =  <. x ,. y >.  ->  z  =  <. u  ,  v >. )  ->  ( z  =  <. x ,  y >.  ->  E. v z  =  <. u ,  v >. ) )
14:12,13:  |-  ( x  =  u  ->  ( z  =  <. x ,. y >.  ->  E. v  z  =  <. u ,  v >. ) )
15:14:  |-  ( E. u x  =  u  ->  E. u ( z  =  <. x ,. y  >.  ->  E. v z  =  <. u ,  v >. ) )
150::  |-  ( u  =  x  <->  x  =  u )
151:150:  |-  ( E. u u  =  x  <->  E. u x  =  u )
152::  |-  E. u u  =  x
16:151,152:  |-  E. u x  =  u
17:15,16:  |-  E. u ( z  =  <. x ,. y >.  ->  E. v z  =  <.  u ,  v >. )
18:17:  |-  ( z  =  <. x ,. y >.  ->  E. u E. v z  =  <.  u ,  v >. )
19:18:  |-  ( E. y z  =  <. x ,. y >.  ->  E. y E. u  E. v z  =  <. u ,  v >. )
20::  |-  ( E. y E. u E. v z  =  <. u ,. v >.  ->  E. u E. v z  =  <. u ,  v >. )
21:19,20:  |-  ( E. y z  =  <. x ,. y >.  ->  E. u E. v z  =  <. u ,  v >. )
22:21:  |-  ( E. x E. y z  =  <. x ,. y >.  ->  E. x  E. u E. v z  =  <. u ,  v >. )
23::  |-  ( E. x E. u E. v z  =  <. u ,. v >.  ->  E. u E. v z  =  <. u ,  v >. )
24:22,23:  |-  ( E. x E. y z  =  <. x ,. y >.  ->  E. u  E. v z  =  <. u ,  v >. )
25:24:  |-  { z  |  E. x E. y z  =  <. x ,. y >. }  C_  { z  |  E. u E. v z  =  <. u ,  v >. }
26::  |-  x  e.  _V
27::  |-  y  e.  _V
28:26,27:  |-  ( x  e.  _V  /\  y  e.  _V )
29:28:  |-  ( z  =  <. x ,. y >.  <->  ( z  =  <. x ,. y  >.  /\  ( x  e.  _V  /\  y  e.  _V ) ) )
30:29:  |-  ( E. y z  =  <. x ,. y >.  <->  E. y ( z  =  <. x ,  y >.  /\  ( x  e.  _V  /\  y  e.  _V ) ) )
31:30:  |-  ( E. x E. y z  =  <. x ,. y >.  <->  E. x  E. y ( z  =  <. x ,  y >.  /\  ( x  e.  _V  /\  y  e.  _V ) ) )
32:31:  |-  { z  |  E. x E. y z  =  <. x ,. y >. }  =  {  z  |  E. x E. y ( z  =  <. x ,  y >.  /\  ( x  e.  _V  /\  y  e.  _V ) ) }
320:25,32:  |-  { z  |  E. x E. y ( z  =  <. x ,. y >.  /\  ( x  e.  _V  /\  y  e.  _V ) ) }  C_  { z  |  E. u E. v z  =  <. u ,  v >. }
33::  |-  u  e.  _V
34::  |-  v  e.  _V
35:33,34:  |-  ( u  e.  _V  /\  v  e.  _V )
36:35:  |-  ( z  =  <. u ,. v >.  <->  ( z  =  <. u ,. v  >.  /\  ( u  e.  _V  /\  v  e.  _V ) ) )
37:36:  |-  ( E. v z  =  <. u ,. v >.  <->  E. v ( z  =  <. u ,  v >.  /\  ( u  e.  _V  /\  v  e.  _V ) ) )
38:37:  |-  ( E. u E. v z  =  <. u ,. v >.  <->  E. u  E. v ( z  =  <. u ,  v >.  /\  ( u  e.  _V  /\  v  e.  _V ) ) )
39:38:  |-  { z  |  E. u E. v z  =  <. u ,. v >. }  =  {  z  |  E. u E. v ( z  =  <. u ,  v >.  /\  ( u  e.  _V  /\  v  e.  _V ) ) }
40:320,39:  |-  { z  |  E. x E. y ( z  =  <. x ,. y >.  /\  ( x  e.  _V  /\  y  e.  _V ) ) }  C_  { z  |  E. u E. v ( z  =  <. u ,  v >.  /\  ( u  e.  _V  /\  v  e.  _V ) ) }
41::  |-  { <. x ,. y >.  |  ( x  e.  _V  /\  y  e.  _V  ) }  =  { z  |  E. x E. y ( z  =  <. x ,  y >.  /\  ( x  e.  _V  /\  y  e.  _V ) )  }
42::  |-  { <. u ,. v >.  |  ( u  e.  _V  /\  v  e.  _V  ) }  =  { z  |  E. u E. v ( z  =  <. u ,  v >.  /\  ( u  e.  _V  /\  v  e.  _V ) )  }
43:40,41,42:  |-  { <. x ,. y >.  |  ( x  e.  _V  /\  y  e.  _V  ) }  C_  { <. u ,  v >.  |  ( u  e.  _V  /\  v  e.  _V ) }
44::  |-  { <. u ,. v >.  |  ( u  e.  _V  /\  v  e.  _V  ) }  =  ( _V  X.  _V )
45:43,44:  |-  { <. x ,. y >.  |  ( x  e.  _V  /\  y  e.  _V  ) }  C_  ( _V  X.  _V )
46:28:  |-  ( ph  ->  ( x  e.  _V  /\  y  e.  _V ) )
47:46:  |-  { <. x ,. y >.  |  ph }  C_  { <. x ,. y >.  |  ( x  e.  _V  /\  y  e.  _V ) }
48:45,47:  |-  { <. x ,. y >.  |  ph }  C_  ( _V  X.  _V )
qed:48:  |-  Rel  { <. x ,. y >.  |  ph }
(Contributed by Alan Sare, 9-Jul-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
relopabVD  |-  Rel  { <. x ,  y >.  |  ph }

Proof of Theorem relopabVD
Dummy variables  z 
v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2951 . . . . . 6  |-  x  e. 
_V
2 vex 2951 . . . . . 6  |-  y  e. 
_V
31, 2pm3.2i 442 . . . . 5  |-  ( x  e.  _V  /\  y  e.  _V )
43a1i 11 . . . 4  |-  ( ph  ->  ( x  e.  _V  /\  y  e.  _V )
)
54ssopab2i 4474 . . 3  |-  { <. x ,  y >.  |  ph }  C_  { <. x ,  y >.  |  ( x  e.  _V  /\  y  e.  _V ) }
63biantru 492 . . . . . . . . . 10  |-  ( z  =  <. x ,  y
>. 
<->  ( z  =  <. x ,  y >.  /\  (
x  e.  _V  /\  y  e.  _V )
) )
76exbii 1592 . . . . . . . . 9  |-  ( E. y  z  =  <. x ,  y >.  <->  E. y
( z  =  <. x ,  y >.  /\  (
x  e.  _V  /\  y  e.  _V )
) )
87exbii 1592 . . . . . . . 8  |-  ( E. x E. y  z  =  <. x ,  y
>. 
<->  E. x E. y
( z  =  <. x ,  y >.  /\  (
x  e.  _V  /\  y  e.  _V )
) )
98abbii 2547 . . . . . . 7  |-  { z  |  E. x E. y  z  =  <. x ,  y >. }  =  { z  |  E. x E. y ( z  =  <. x ,  y
>.  /\  ( x  e. 
_V  /\  y  e.  _V ) ) }
10 a9ev 1668 . . . . . . . . . . . . . . 15  |-  E. u  u  =  x
11 equcom 1692 . . . . . . . . . . . . . . . 16  |-  ( u  =  x  <->  x  =  u )
1211exbii 1592 . . . . . . . . . . . . . . 15  |-  ( E. u  u  =  x  <->  E. u  x  =  u )
1310, 12mpbi 200 . . . . . . . . . . . . . 14  |-  E. u  x  =  u
14 a9ev 1668 . . . . . . . . . . . . . . . . . . 19  |-  E. v 
v  =  y
15 equcom 1692 . . . . . . . . . . . . . . . . . . . 20  |-  ( v  =  y  <->  y  =  v )
1615exbii 1592 . . . . . . . . . . . . . . . . . . 19  |-  ( E. v  v  =  y  <->  E. v  y  =  v )
1714, 16mpbi 200 . . . . . . . . . . . . . . . . . 18  |-  E. v 
y  =  v
18 idn1 28602 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  (. y  =  v  ->.  y  =  v ).
19 opeq2 3977 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( y  =  v  ->  <. x ,  y >.  =  <. x ,  v >. )
2018, 19e1_ 28665 . . . . . . . . . . . . . . . . . . . . . . 23  |-  (. y  =  v  ->.  <. x ,  y
>.  =  <. x ,  v >. ).
21 idn2 28651 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  (. y  =  v ,. x  =  u  ->.  x  =  u ).
22 opeq1 3976 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( x  =  u  ->  <. x ,  v >.  =  <. u ,  v >. )
2321, 22e2 28669 . . . . . . . . . . . . . . . . . . . . . . 23  |-  (. y  =  v ,. x  =  u  ->.  <. x ,  v
>.  =  <. u ,  v >. ).
24 eqeq1 2441 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( <.
x ,  y >.  =  <. x ,  v
>.  ->  ( <. x ,  y >.  =  <. u ,  v >.  <->  <. x ,  v >.  =  <. u ,  v >. )
)
2524biimprd 215 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( <.
x ,  y >.  =  <. x ,  v
>.  ->  ( <. x ,  v >.  =  <. u ,  v >.  ->  <. x ,  y >.  =  <. u ,  v >. )
)
2620, 23, 25e12 28773 . . . . . . . . . . . . . . . . . . . . . 22  |-  (. y  =  v ,. x  =  u  ->.  <. x ,  y
>.  =  <. u ,  v >. ).
27 eqeq2 2444 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( <.
x ,  y >.  =  <. u ,  v
>.  ->  ( z  = 
<. x ,  y >.  <->  z  =  <. u ,  v
>. ) )
2827biimpd 199 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( <.
x ,  y >.  =  <. u ,  v
>.  ->  ( z  = 
<. x ,  y >.  ->  z  =  <. u ,  v >. )
)
2926, 28e2 28669 . . . . . . . . . . . . . . . . . . . . 21  |-  (. y  =  v ,. x  =  u  ->.  ( z  = 
<. x ,  y >.  ->  z  =  <. u ,  v >. ) ).
3029in2 28643 . . . . . . . . . . . . . . . . . . . 20  |-  (. y  =  v  ->.  ( x  =  u  ->  ( z  =  <. x ,  y
>.  ->  z  =  <. u ,  v >. )
) ).
3130in1 28599 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  v  ->  (
x  =  u  -> 
( z  =  <. x ,  y >.  ->  z  =  <. u ,  v
>. ) ) )
3231eximi 1585 . . . . . . . . . . . . . . . . . 18  |-  ( E. v  y  =  v  ->  E. v ( x  =  u  ->  (
z  =  <. x ,  y >.  ->  z  =  <. u ,  v
>. ) ) )
3317, 32ax-mp 8 . . . . . . . . . . . . . . . . 17  |-  E. v
( x  =  u  ->  ( z  = 
<. x ,  y >.  ->  z  =  <. u ,  v >. )
)
343319.37aiv 1923 . . . . . . . . . . . . . . . 16  |-  ( x  =  u  ->  E. v
( z  =  <. x ,  y >.  ->  z  =  <. u ,  v
>. ) )
35 19.37v 1922 . . . . . . . . . . . . . . . . 17  |-  ( E. v ( z  = 
<. x ,  y >.  ->  z  =  <. u ,  v >. )  <->  ( z  =  <. x ,  y >.  ->  E. v 
z  =  <. u ,  v >. )
)
3635biimpi 187 . . . . . . . . . . . . . . . 16  |-  ( E. v ( z  = 
<. x ,  y >.  ->  z  =  <. u ,  v >. )  ->  ( z  =  <. x ,  y >.  ->  E. v 
z  =  <. u ,  v >. )
)
3734, 36syl 16 . . . . . . . . . . . . . . 15  |-  ( x  =  u  ->  (
z  =  <. x ,  y >.  ->  E. v 
z  =  <. u ,  v >. )
)
3837eximi 1585 . . . . . . . . . . . . . 14  |-  ( E. u  x  =  u  ->  E. u ( z  =  <. x ,  y
>.  ->  E. v  z  = 
<. u ,  v >.
) )
3913, 38ax-mp 8 . . . . . . . . . . . . 13  |-  E. u
( z  =  <. x ,  y >.  ->  E. v 
z  =  <. u ,  v >. )
403919.37aiv 1923 . . . . . . . . . . . 12  |-  ( z  =  <. x ,  y
>.  ->  E. u E. v 
z  =  <. u ,  v >. )
4140eximi 1585 . . . . . . . . . . 11  |-  ( E. y  z  =  <. x ,  y >.  ->  E. y E. u E. v  z  =  <. u ,  v
>. )
42 19.9v 1676 . . . . . . . . . . . 12  |-  ( E. y E. u E. v  z  =  <. u ,  v >.  <->  E. u E. v  z  =  <. u ,  v >.
)
4342biimpi 187 . . . . . . . . . . 11  |-  ( E. y E. u E. v  z  =  <. u ,  v >.  ->  E. u E. v  z  =  <. u ,  v >.
)
4441, 43syl 16 . . . . . . . . . 10  |-  ( E. y  z  =  <. x ,  y >.  ->  E. u E. v  z  =  <. u ,  v >.
)
4544eximi 1585 . . . . . . . . 9  |-  ( E. x E. y  z  =  <. x ,  y
>.  ->  E. x E. u E. v  z  =  <. u ,  v >.
)
46 19.9v 1676 . . . . . . . . . 10  |-  ( E. x E. u E. v  z  =  <. u ,  v >.  <->  E. u E. v  z  =  <. u ,  v >.
)
4746biimpi 187 . . . . . . . . 9  |-  ( E. x E. u E. v  z  =  <. u ,  v >.  ->  E. u E. v  z  =  <. u ,  v >.
)
4845, 47syl 16 . . . . . . . 8  |-  ( E. x E. y  z  =  <. x ,  y
>.  ->  E. u E. v 
z  =  <. u ,  v >. )
4948ss2abi 3407 . . . . . . 7  |-  { z  |  E. x E. y  z  =  <. x ,  y >. }  C_  { z  |  E. u E. v  z  =  <. u ,  v >. }
509, 49eqsstr3i 3371 . . . . . 6  |-  { z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  ( x  e.  _V  /\  y  e.  _V )
) }  C_  { z  |  E. u E. v  z  =  <. u ,  v >. }
51 vex 2951 . . . . . . . . . . 11  |-  u  e. 
_V
52 vex 2951 . . . . . . . . . . 11  |-  v  e. 
_V
5351, 52pm3.2i 442 . . . . . . . . . 10  |-  ( u  e.  _V  /\  v  e.  _V )
5453biantru 492 . . . . . . . . 9  |-  ( z  =  <. u ,  v
>. 
<->  ( z  =  <. u ,  v >.  /\  (
u  e.  _V  /\  v  e.  _V )
) )
5554exbii 1592 . . . . . . . 8  |-  ( E. v  z  =  <. u ,  v >.  <->  E. v
( z  =  <. u ,  v >.  /\  (
u  e.  _V  /\  v  e.  _V )
) )
5655exbii 1592 . . . . . . 7  |-  ( E. u E. v  z  =  <. u ,  v
>. 
<->  E. u E. v
( z  =  <. u ,  v >.  /\  (
u  e.  _V  /\  v  e.  _V )
) )
5756abbii 2547 . . . . . 6  |-  { z  |  E. u E. v  z  =  <. u ,  v >. }  =  { z  |  E. u E. v ( z  =  <. u ,  v
>.  /\  ( u  e. 
_V  /\  v  e.  _V ) ) }
5850, 57sseqtri 3372 . . . . 5  |-  { z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  ( x  e.  _V  /\  y  e.  _V )
) }  C_  { z  |  E. u E. v ( z  = 
<. u ,  v >.  /\  ( u  e.  _V  /\  v  e.  _V )
) }
59 df-opab 4259 . . . . 5  |-  { <. x ,  y >.  |  ( x  e.  _V  /\  y  e.  _V ) }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  (
x  e.  _V  /\  y  e.  _V )
) }
60 df-opab 4259 . . . . 5  |-  { <. u ,  v >.  |  ( u  e.  _V  /\  v  e.  _V ) }  =  { z  |  E. u E. v
( z  =  <. u ,  v >.  /\  (
u  e.  _V  /\  v  e.  _V )
) }
6158, 59, 603sstr4i 3379 . . . 4  |-  { <. x ,  y >.  |  ( x  e.  _V  /\  y  e.  _V ) }  C_  { <. u ,  v >.  |  ( u  e.  _V  /\  v  e.  _V ) }
62 df-xp 4876 . . . . 5  |-  ( _V 
X.  _V )  =  { <. u ,  v >.  |  ( u  e. 
_V  /\  v  e.  _V ) }
6362eqcomi 2439 . . . 4  |-  { <. u ,  v >.  |  ( u  e.  _V  /\  v  e.  _V ) }  =  ( _V  X.  _V )
6461, 63sseqtri 3372 . . 3  |-  { <. x ,  y >.  |  ( x  e.  _V  /\  y  e.  _V ) }  C_  ( _V  X.  _V )
655, 64sstri 3349 . 2  |-  { <. x ,  y >.  |  ph }  C_  ( _V  X.  _V )
66 df-rel 4877 . . 3  |-  ( Rel 
{ <. x ,  y
>.  |  ph }  <->  { <. x ,  y >.  |  ph }  C_  ( _V  X.  _V ) )
6766biimpri 198 . 2  |-  ( {
<. x ,  y >.  |  ph }  C_  ( _V  X.  _V )  ->  Rel  { <. x ,  y
>.  |  ph } )
6865, 67e0_ 28821 1  |-  Rel  { <. x ,  y >.  |  ph }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2421   _Vcvv 2948    C_ wss 3312   <.cop 3809   {copab 4257    X. cxp 4868   Rel wrel 4875
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
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-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  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-opab 4259  df-xp 4876  df-rel 4877  df-vd1 28598  df-vd2 28607
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