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Theorem relopabVD 28993
Description: Virtual deduction proof of relopab 4828. 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 4828 is relopabVD 28993 without virtual deductions and was automatically derived from relopabVD 28993.
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 2804 . . . . . 6  |-  x  e. 
_V
2 vex 2804 . . . . . 6  |-  y  e. 
_V
31, 2pm3.2i 441 . . . . 5  |-  ( x  e.  _V  /\  y  e.  _V )
43a1i 10 . . . 4  |-  ( ph  ->  ( x  e.  _V  /\  y  e.  _V )
)
54ssopab2i 4308 . . 3  |-  { <. x ,  y >.  |  ph }  C_  { <. x ,  y >.  |  ( x  e.  _V  /\  y  e.  _V ) }
63biantru 491 . . . . . . . . . 10  |-  ( z  =  <. x ,  y
>. 
<->  ( z  =  <. x ,  y >.  /\  (
x  e.  _V  /\  y  e.  _V )
) )
76exbii 1572 . . . . . . . . 9  |-  ( E. y  z  =  <. x ,  y >.  <->  E. y
( z  =  <. x ,  y >.  /\  (
x  e.  _V  /\  y  e.  _V )
) )
87exbii 1572 . . . . . . . 8  |-  ( E. x E. y  z  =  <. x ,  y
>. 
<->  E. x E. y
( z  =  <. x ,  y >.  /\  (
x  e.  _V  /\  y  e.  _V )
) )
98abbii 2408 . . . . . . 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 1646 . . . . . . . . . . . . . . 15  |-  E. u  u  =  x
11 equcom 1665 . . . . . . . . . . . . . . . 16  |-  ( u  =  x  <->  x  =  u )
1211exbii 1572 . . . . . . . . . . . . . . 15  |-  ( E. u  u  =  x  <->  E. u  x  =  u )
1310, 12mpbi 199 . . . . . . . . . . . . . 14  |-  E. u  x  =  u
14 a9ev 1646 . . . . . . . . . . . . . . . . . . 19  |-  E. v 
v  =  y
15 equcom 1665 . . . . . . . . . . . . . . . . . . . 20  |-  ( v  =  y  <->  y  =  v )
1615exbii 1572 . . . . . . . . . . . . . . . . . . 19  |-  ( E. v  v  =  y  <->  E. v  y  =  v )
1714, 16mpbi 199 . . . . . . . . . . . . . . . . . 18  |-  E. v 
y  =  v
18 idn1 28641 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  (. y  =  v  ->.  y  =  v ).
19 opeq2 3813 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( y  =  v  ->  <. x ,  y >.  =  <. x ,  v >. )
2018, 19e1_ 28704 . . . . . . . . . . . . . . . . . . . . . . 23  |-  (. y  =  v  ->.  <. x ,  y
>.  =  <. x ,  v >. ).
21 idn2 28690 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  (. y  =  v ,. x  =  u  ->.  x  =  u ).
22 opeq1 3812 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( x  =  u  ->  <. x ,  v >.  =  <. u ,  v >. )
2321, 22e2 28708 . . . . . . . . . . . . . . . . . . . . . . 23  |-  (. y  =  v ,. x  =  u  ->.  <. x ,  v
>.  =  <. u ,  v >. ).
24 eqeq1 2302 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( <.
x ,  y >.  =  <. x ,  v
>.  ->  ( <. x ,  y >.  =  <. u ,  v >.  <->  <. x ,  v >.  =  <. u ,  v >. )
)
2524biimprd 214 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( <.
x ,  y >.  =  <. x ,  v
>.  ->  ( <. x ,  v >.  =  <. u ,  v >.  ->  <. x ,  y >.  =  <. u ,  v >. )
)
2620, 23, 25e12 28813 . . . . . . . . . . . . . . . . . . . . . 22  |-  (. y  =  v ,. x  =  u  ->.  <. x ,  y
>.  =  <. u ,  v >. ).
27 eqeq2 2305 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( <.
x ,  y >.  =  <. u ,  v
>.  ->  ( z  = 
<. x ,  y >.  <->  z  =  <. u ,  v
>. ) )
2827biimpd 198 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( <.
x ,  y >.  =  <. u ,  v
>.  ->  ( z  = 
<. x ,  y >.  ->  z  =  <. u ,  v >. )
)
2926, 28e2 28708 . . . . . . . . . . . . . . . . . . . . 21  |-  (. y  =  v ,. x  =  u  ->.  ( z  = 
<. x ,  y >.  ->  z  =  <. u ,  v >. ) ).
3029in2 28682 . . . . . . . . . . . . . . . . . . . 20  |-  (. y  =  v  ->.  ( x  =  u  ->  ( z  =  <. x ,  y
>.  ->  z  =  <. u ,  v >. )
) ).
3130in1 28638 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  v  ->  (
x  =  u  -> 
( z  =  <. x ,  y >.  ->  z  =  <. u ,  v
>. ) ) )
3231eximi 1566 . . . . . . . . . . . . . . . . . 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 1853 . . . . . . . . . . . . . . . 16  |-  ( x  =  u  ->  E. v
( z  =  <. x ,  y >.  ->  z  =  <. u ,  v
>. ) )
35 19.37v 1852 . . . . . . . . . . . . . . . . 17  |-  ( E. v ( z  = 
<. x ,  y >.  ->  z  =  <. u ,  v >. )  <->  ( z  =  <. x ,  y >.  ->  E. v 
z  =  <. u ,  v >. )
)
3635biimpi 186 . . . . . . . . . . . . . . . 16  |-  ( E. v ( z  = 
<. x ,  y >.  ->  z  =  <. u ,  v >. )  ->  ( z  =  <. x ,  y >.  ->  E. v 
z  =  <. u ,  v >. )
)
3734, 36syl 15 . . . . . . . . . . . . . . 15  |-  ( x  =  u  ->  (
z  =  <. x ,  y >.  ->  E. v 
z  =  <. u ,  v >. )
)
3837eximi 1566 . . . . . . . . . . . . . 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 1853 . . . . . . . . . . . 12  |-  ( z  =  <. x ,  y
>.  ->  E. u E. v 
z  =  <. u ,  v >. )
4140eximi 1566 . . . . . . . . . . 11  |-  ( E. y  z  =  <. x ,  y >.  ->  E. y E. u E. v  z  =  <. u ,  v
>. )
42 19.9v 1653 . . . . . . . . . . . 12  |-  ( E. y E. u E. v  z  =  <. u ,  v >.  <->  E. u E. v  z  =  <. u ,  v >.
)
4342biimpi 186 . . . . . . . . . . 11  |-  ( E. y E. u E. v  z  =  <. u ,  v >.  ->  E. u E. v  z  =  <. u ,  v >.
)
4441, 43syl 15 . . . . . . . . . 10  |-  ( E. y  z  =  <. x ,  y >.  ->  E. u E. v  z  =  <. u ,  v >.
)
4544eximi 1566 . . . . . . . . 9  |-  ( E. x E. y  z  =  <. x ,  y
>.  ->  E. x E. u E. v  z  =  <. u ,  v >.
)
46 19.9v 1653 . . . . . . . . . 10  |-  ( E. x E. u E. v  z  =  <. u ,  v >.  <->  E. u E. v  z  =  <. u ,  v >.
)
4746biimpi 186 . . . . . . . . 9  |-  ( E. x E. u E. v  z  =  <. u ,  v >.  ->  E. u E. v  z  =  <. u ,  v >.
)
4845, 47syl 15 . . . . . . . 8  |-  ( E. x E. y  z  =  <. x ,  y
>.  ->  E. u E. v 
z  =  <. u ,  v >. )
4948ss2abi 3258 . . . . . . 7  |-  { z  |  E. x E. y  z  =  <. x ,  y >. }  C_  { z  |  E. u E. v  z  =  <. u ,  v >. }
509, 49eqsstr3i 3222 . . . . . 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 2804 . . . . . . . . . . 11  |-  u  e. 
_V
52 vex 2804 . . . . . . . . . . 11  |-  v  e. 
_V
5351, 52pm3.2i 441 . . . . . . . . . 10  |-  ( u  e.  _V  /\  v  e.  _V )
5453biantru 491 . . . . . . . . 9  |-  ( z  =  <. u ,  v
>. 
<->  ( z  =  <. u ,  v >.  /\  (
u  e.  _V  /\  v  e.  _V )
) )
5554exbii 1572 . . . . . . . 8  |-  ( E. v  z  =  <. u ,  v >.  <->  E. v
( z  =  <. u ,  v >.  /\  (
u  e.  _V  /\  v  e.  _V )
) )
5655exbii 1572 . . . . . . 7  |-  ( E. u E. v  z  =  <. u ,  v
>. 
<->  E. u E. v
( z  =  <. u ,  v >.  /\  (
u  e.  _V  /\  v  e.  _V )
) )
5756abbii 2408 . . . . . 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 3223 . . . . 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 4094 . . . . 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 4094 . . . . 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 3230 . . . 4  |-  { <. x ,  y >.  |  ( x  e.  _V  /\  y  e.  _V ) }  C_  { <. u ,  v >.  |  ( u  e.  _V  /\  v  e.  _V ) }
62 df-xp 4711 . . . . 5  |-  ( _V 
X.  _V )  =  { <. u ,  v >.  |  ( u  e. 
_V  /\  v  e.  _V ) }
6362eqcomi 2300 . . . 4  |-  { <. u ,  v >.  |  ( u  e.  _V  /\  v  e.  _V ) }  =  ( _V  X.  _V )
6461, 63sseqtri 3223 . . 3  |-  { <. x ,  y >.  |  ( x  e.  _V  /\  y  e.  _V ) }  C_  ( _V  X.  _V )
655, 64sstri 3201 . 2  |-  { <. x ,  y >.  |  ph }  C_  ( _V  X.  _V )
66 df-rel 4712 . . 3  |-  ( Rel 
{ <. x ,  y
>.  |  ph }  <->  { <. x ,  y >.  |  ph }  C_  ( _V  X.  _V ) )
6766biimpri 197 . 2  |-  ( {
<. x ,  y >.  |  ph }  C_  ( _V  X.  _V )  ->  Rel  { <. x ,  y
>.  |  ph } )
6865, 67e0_ 28861 1  |-  Rel  { <. x ,  y >.  |  ph }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282   _Vcvv 2801    C_ wss 3165   <.cop 3656   {copab 4092    X. cxp 4703   Rel wrel 4710
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-opab 4094  df-xp 4711  df-rel 4712  df-vd1 28637  df-vd2 28646
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