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Theorem relopabVD 28354
Description: Virtual deduction proof of relopab 4941. 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 4941 is relopabVD 28354 without virtual deductions and was automatically derived from relopabVD 28354.
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 2902 . . . . . 6  |-  x  e. 
_V
2 vex 2902 . . . . . 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 4423 . . 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 1589 . . . . . . . . 9  |-  ( E. y  z  =  <. x ,  y >.  <->  E. y
( z  =  <. x ,  y >.  /\  (
x  e.  _V  /\  y  e.  _V )
) )
87exbii 1589 . . . . . . . 8  |-  ( E. x E. y  z  =  <. x ,  y
>. 
<->  E. x E. y
( z  =  <. x ,  y >.  /\  (
x  e.  _V  /\  y  e.  _V )
) )
98abbii 2499 . . . . . . 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 1663 . . . . . . . . . . . . . . 15  |-  E. u  u  =  x
11 equcom 1687 . . . . . . . . . . . . . . . 16  |-  ( u  =  x  <->  x  =  u )
1211exbii 1589 . . . . . . . . . . . . . . 15  |-  ( E. u  u  =  x  <->  E. u  x  =  u )
1310, 12mpbi 200 . . . . . . . . . . . . . 14  |-  E. u  x  =  u
14 a9ev 1663 . . . . . . . . . . . . . . . . . . 19  |-  E. v 
v  =  y
15 equcom 1687 . . . . . . . . . . . . . . . . . . . 20  |-  ( v  =  y  <->  y  =  v )
1615exbii 1589 . . . . . . . . . . . . . . . . . . 19  |-  ( E. v  v  =  y  <->  E. v  y  =  v )
1714, 16mpbi 200 . . . . . . . . . . . . . . . . . 18  |-  E. v 
y  =  v
18 idn1 28006 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  (. y  =  v  ->.  y  =  v ).
19 opeq2 3927 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( y  =  v  ->  <. x ,  y >.  =  <. x ,  v >. )
2018, 19e1_ 28069 . . . . . . . . . . . . . . . . . . . . . . 23  |-  (. y  =  v  ->.  <. x ,  y
>.  =  <. x ,  v >. ).
21 idn2 28055 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  (. y  =  v ,. x  =  u  ->.  x  =  u ).
22 opeq1 3926 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( x  =  u  ->  <. x ,  v >.  =  <. u ,  v >. )
2321, 22e2 28073 . . . . . . . . . . . . . . . . . . . . . . 23  |-  (. y  =  v ,. x  =  u  ->.  <. x ,  v
>.  =  <. u ,  v >. ).
24 eqeq1 2393 . . . . . . . . . . . . . . . . . . . . . . . 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 28177 . . . . . . . . . . . . . . . . . . . . . 22  |-  (. y  =  v ,. x  =  u  ->.  <. x ,  y
>.  =  <. u ,  v >. ).
27 eqeq2 2396 . . . . . . . . . . . . . . . . . . . . . . 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 28073 . . . . . . . . . . . . . . . . . . . . 21  |-  (. y  =  v ,. x  =  u  ->.  ( z  = 
<. x ,  y >.  ->  z  =  <. u ,  v >. ) ).
3029in2 28047 . . . . . . . . . . . . . . . . . . . 20  |-  (. y  =  v  ->.  ( x  =  u  ->  ( z  =  <. x ,  y
>.  ->  z  =  <. u ,  v >. )
) ).
3130in1 28003 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  v  ->  (
x  =  u  -> 
( z  =  <. x ,  y >.  ->  z  =  <. u ,  v
>. ) ) )
3231eximi 1582 . . . . . . . . . . . . . . . . . 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 1912 . . . . . . . . . . . . . . . 16  |-  ( x  =  u  ->  E. v
( z  =  <. x ,  y >.  ->  z  =  <. u ,  v
>. ) )
35 19.37v 1911 . . . . . . . . . . . . . . . . 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 1582 . . . . . . . . . . . . . 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 1912 . . . . . . . . . . . 12  |-  ( z  =  <. x ,  y
>.  ->  E. u E. v 
z  =  <. u ,  v >. )
4140eximi 1582 . . . . . . . . . . 11  |-  ( E. y  z  =  <. x ,  y >.  ->  E. y E. u E. v  z  =  <. u ,  v
>. )
42 19.9v 1671 . . . . . . . . . . . 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 1582 . . . . . . . . 9  |-  ( E. x E. y  z  =  <. x ,  y
>.  ->  E. x E. u E. v  z  =  <. u ,  v >.
)
46 19.9v 1671 . . . . . . . . . 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 3358 . . . . . . 7  |-  { z  |  E. x E. y  z  =  <. x ,  y >. }  C_  { z  |  E. u E. v  z  =  <. u ,  v >. }
509, 49eqsstr3i 3322 . . . . . 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 2902 . . . . . . . . . . 11  |-  u  e. 
_V
52 vex 2902 . . . . . . . . . . 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 1589 . . . . . . . 8  |-  ( E. v  z  =  <. u ,  v >.  <->  E. v
( z  =  <. u ,  v >.  /\  (
u  e.  _V  /\  v  e.  _V )
) )
5655exbii 1589 . . . . . . 7  |-  ( E. u E. v  z  =  <. u ,  v
>. 
<->  E. u E. v
( z  =  <. u ,  v >.  /\  (
u  e.  _V  /\  v  e.  _V )
) )
5756abbii 2499 . . . . . 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 3323 . . . . 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 4208 . . . . 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 4208 . . . . 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 3330 . . . 4  |-  { <. x ,  y >.  |  ( x  e.  _V  /\  y  e.  _V ) }  C_  { <. u ,  v >.  |  ( u  e.  _V  /\  v  e.  _V ) }
62 df-xp 4824 . . . . 5  |-  ( _V 
X.  _V )  =  { <. u ,  v >.  |  ( u  e. 
_V  /\  v  e.  _V ) }
6362eqcomi 2391 . . . 4  |-  { <. u ,  v >.  |  ( u  e.  _V  /\  v  e.  _V ) }  =  ( _V  X.  _V )
6461, 63sseqtri 3323 . . 3  |-  { <. x ,  y >.  |  ( x  e.  _V  /\  y  e.  _V ) }  C_  ( _V  X.  _V )
655, 64sstri 3300 . 2  |-  { <. x ,  y >.  |  ph }  C_  ( _V  X.  _V )
66 df-rel 4825 . . 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_ 28225 1  |-  Rel  { <. x ,  y >.  |  ph }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   {cab 2373   _Vcvv 2899    C_ wss 3263   <.cop 3760   {copab 4206    X. cxp 4816   Rel wrel 4823
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-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
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-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-rab 2658  df-v 2901  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-opab 4208  df-xp 4824  df-rel 4825  df-vd1 28002  df-vd2 28011
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