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Theorem oprabid 5803
Description: The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Although this theorem would be useful with a distinct variable constraint between  x,  y, and  z, we use ax-bndl 1486 to eliminate that constraint. (Contributed by Mario Carneiro, 20-Mar-2013.)
Assertion
Ref Expression
oprabid  |-  ( <. <. x ,  y >. ,  z >.  e.  { <. <. x ,  y
>. ,  z >.  | 
ph }  <->  ph )

Proof of Theorem oprabid
Dummy variables  a  r  s  t  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2689 . . . 4  |-  x  e. 
_V
2 vex 2689 . . . 4  |-  y  e. 
_V
31, 2opex 4151 . . 3  |-  <. x ,  y >.  e.  _V
4 vex 2689 . . 3  |-  z  e. 
_V
5 opexg 4150 . . 3  |-  ( (
<. x ,  y >.  e.  _V  /\  z  e. 
_V )  ->  <. <. x ,  y >. ,  z
>.  e.  _V )
63, 4, 5mp2an 422 . 2  |-  <. <. x ,  y >. ,  z
>.  e.  _V
73, 4eqvinop 4165 . . . . 5  |-  ( w  =  <. <. x ,  y
>. ,  z >.  <->  E. a E. t ( w  =  <. a ,  t
>.  /\  <. a ,  t
>.  =  <. <. x ,  y >. ,  z
>. ) )
87biimpi 119 . . . 4  |-  ( w  =  <. <. x ,  y
>. ,  z >.  ->  E. a E. t ( w  =  <. a ,  t >.  /\  <. a ,  t >.  =  <. <.
x ,  y >. ,  z >. )
)
9 eqeq1 2146 . . . . . . . 8  |-  ( w  =  <. a ,  t
>.  ->  ( w  = 
<. <. x ,  y
>. ,  z >.  <->  <. a ,  t >.  =  <. <.
x ,  y >. ,  z >. )
)
10 vex 2689 . . . . . . . . 9  |-  a  e. 
_V
11 vex 2689 . . . . . . . . 9  |-  t  e. 
_V
1210, 11opth1 4158 . . . . . . . 8  |-  ( <.
a ,  t >.  =  <. <. x ,  y
>. ,  z >.  -> 
a  =  <. x ,  y >. )
139, 12syl6bi 162 . . . . . . 7  |-  ( w  =  <. a ,  t
>.  ->  ( w  = 
<. <. x ,  y
>. ,  z >.  -> 
a  =  <. x ,  y >. )
)
141, 2eqvinop 4165 . . . . . . . . 9  |-  ( a  =  <. x ,  y
>. 
<->  E. r E. s
( a  =  <. r ,  s >.  /\  <. r ,  s >.  =  <. x ,  y >. )
)
15 opeq1 3705 . . . . . . . . . . . . 13  |-  ( a  =  <. r ,  s
>.  ->  <. a ,  t
>.  =  <. <. r ,  s >. ,  t
>. )
1615eqeq2d 2151 . . . . . . . . . . . 12  |-  ( a  =  <. r ,  s
>.  ->  ( w  = 
<. a ,  t >.  <->  w  =  <. <. r ,  s
>. ,  t >. ) )
171, 2, 4otth2 4163 . . . . . . . . . . . . . . . . . . 19  |-  ( <. <. x ,  y >. ,  z >.  =  <. <.
r ,  s >. ,  t >.  <->  ( x  =  r  /\  y  =  s  /\  z  =  t ) )
18 df-3an 964 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  =  r  /\  y  =  s  /\  z  =  t )  <->  ( ( x  =  r  /\  y  =  s )  /\  z  =  t ) )
1917, 18bitri 183 . . . . . . . . . . . . . . . . . 18  |-  ( <. <. x ,  y >. ,  z >.  =  <. <.
r ,  s >. ,  t >.  <->  ( (
x  =  r  /\  y  =  s )  /\  z  =  t
) )
2019anbi1i 453 . . . . . . . . . . . . . . . . 17  |-  ( (
<. <. x ,  y
>. ,  z >.  = 
<. <. r ,  s
>. ,  t >.  /\ 
ph )  <->  ( (
( x  =  r  /\  y  =  s )  /\  z  =  t )  /\  ph ) )
21 anass 398 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( x  =  r  /\  y  =  s )  /\  z  =  t )  /\  ph )  <->  ( ( x  =  r  /\  y  =  s )  /\  ( z  =  t  /\  ph ) ) )
22 anass 398 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  =  r  /\  y  =  s )  /\  ( z  =  t  /\  ph ) )  <->  ( x  =  r  /\  (
y  =  s  /\  ( z  =  t  /\  ph ) ) ) )
2320, 21, 223bitri 205 . . . . . . . . . . . . . . . 16  |-  ( (
<. <. x ,  y
>. ,  z >.  = 
<. <. r ,  s
>. ,  t >.  /\ 
ph )  <->  ( x  =  r  /\  (
y  =  s  /\  ( z  =  t  /\  ph ) ) ) )
24233exbii 1586 . . . . . . . . . . . . . . 15  |-  ( E. x E. y E. z ( <. <. x ,  y >. ,  z
>.  =  <. <. r ,  s >. ,  t
>.  /\  ph )  <->  E. x E. y E. z ( x  =  r  /\  ( y  =  s  /\  ( z  =  t  /\  ph )
) ) )
25 oprabidlem 5802 . . . . . . . . . . . . . . . . . 18  |-  ( E. x E. z ( x  =  r  /\  ( y  =  s  /\  ( z  =  t  /\  ph )
) )  ->  E. x
( x  =  r  /\  E. z ( y  =  s  /\  ( z  =  t  /\  ph ) ) ) )
2625eximi 1579 . . . . . . . . . . . . . . . . 17  |-  ( E. y E. x E. z ( x  =  r  /\  ( y  =  s  /\  (
z  =  t  /\  ph ) ) )  ->  E. y E. x ( x  =  r  /\  E. z ( y  =  s  /\  ( z  =  t  /\  ph ) ) ) )
27 excom 1642 . . . . . . . . . . . . . . . . 17  |-  ( E. x E. y E. z ( x  =  r  /\  ( y  =  s  /\  (
z  =  t  /\  ph ) ) )  <->  E. y E. x E. z ( x  =  r  /\  ( y  =  s  /\  ( z  =  t  /\  ph )
) ) )
28 excom 1642 . . . . . . . . . . . . . . . . 17  |-  ( E. x E. y ( x  =  r  /\  E. z ( y  =  s  /\  ( z  =  t  /\  ph ) ) )  <->  E. y E. x ( x  =  r  /\  E. z
( y  =  s  /\  ( z  =  t  /\  ph )
) ) )
2926, 27, 283imtr4i 200 . . . . . . . . . . . . . . . 16  |-  ( E. x E. y E. z ( x  =  r  /\  ( y  =  s  /\  (
z  =  t  /\  ph ) ) )  ->  E. x E. y ( x  =  r  /\  E. z ( y  =  s  /\  ( z  =  t  /\  ph ) ) ) )
30 oprabidlem 5802 . . . . . . . . . . . . . . . 16  |-  ( E. x E. y ( x  =  r  /\  E. z ( y  =  s  /\  ( z  =  t  /\  ph ) ) )  ->  E. x ( x  =  r  /\  E. y E. z ( y  =  s  /\  ( z  =  t  /\  ph ) ) ) )
31 oprabidlem 5802 . . . . . . . . . . . . . . . . . 18  |-  ( E. y E. z ( y  =  s  /\  ( z  =  t  /\  ph ) )  ->  E. y ( y  =  s  /\  E. z ( z  =  t  /\  ph )
) )
3231anim2i 339 . . . . . . . . . . . . . . . . 17  |-  ( ( x  =  r  /\  E. y E. z ( y  =  s  /\  ( z  =  t  /\  ph ) ) )  ->  ( x  =  r  /\  E. y
( y  =  s  /\  E. z ( z  =  t  /\  ph ) ) ) )
3332eximi 1579 . . . . . . . . . . . . . . . 16  |-  ( E. x ( x  =  r  /\  E. y E. z ( y  =  s  /\  ( z  =  t  /\  ph ) ) )  ->  E. x ( x  =  r  /\  E. y
( y  =  s  /\  E. z ( z  =  t  /\  ph ) ) ) )
3429, 30, 333syl 17 . . . . . . . . . . . . . . 15  |-  ( E. x E. y E. z ( x  =  r  /\  ( y  =  s  /\  (
z  =  t  /\  ph ) ) )  ->  E. x ( x  =  r  /\  E. y
( y  =  s  /\  E. z ( z  =  t  /\  ph ) ) ) )
3524, 34sylbi 120 . . . . . . . . . . . . . 14  |-  ( E. x E. y E. z ( <. <. x ,  y >. ,  z
>.  =  <. <. r ,  s >. ,  t
>.  /\  ph )  ->  E. x ( x  =  r  /\  E. y
( y  =  s  /\  E. z ( z  =  t  /\  ph ) ) ) )
36 euequ1 2094 . . . . . . . . . . . . . . . . . . 19  |-  E! x  x  =  r
37 eupick 2078 . . . . . . . . . . . . . . . . . . 19  |-  ( ( E! x  x  =  r  /\  E. x
( x  =  r  /\  E. y ( y  =  s  /\  E. z ( z  =  t  /\  ph )
) ) )  -> 
( x  =  r  ->  E. y ( y  =  s  /\  E. z ( z  =  t  /\  ph )
) ) )
3836, 37mpan 420 . . . . . . . . . . . . . . . . . 18  |-  ( E. x ( x  =  r  /\  E. y
( y  =  s  /\  E. z ( z  =  t  /\  ph ) ) )  -> 
( x  =  r  ->  E. y ( y  =  s  /\  E. z ( z  =  t  /\  ph )
) ) )
39 euequ1 2094 . . . . . . . . . . . . . . . . . . . 20  |-  E! y  y  =  s
40 eupick 2078 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E! y  y  =  s  /\  E. y
( y  =  s  /\  E. z ( z  =  t  /\  ph ) ) )  -> 
( y  =  s  ->  E. z ( z  =  t  /\  ph ) ) )
4139, 40mpan 420 . . . . . . . . . . . . . . . . . . 19  |-  ( E. y ( y  =  s  /\  E. z
( z  =  t  /\  ph ) )  ->  ( y  =  s  ->  E. z
( z  =  t  /\  ph ) ) )
42 euequ1 2094 . . . . . . . . . . . . . . . . . . . 20  |-  E! z  z  =  t
43 eupick 2078 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E! z  z  =  t  /\  E. z
( z  =  t  /\  ph ) )  ->  ( z  =  t  ->  ph ) )
4442, 43mpan 420 . . . . . . . . . . . . . . . . . . 19  |-  ( E. z ( z  =  t  /\  ph )  ->  ( z  =  t  ->  ph ) )
4541, 44syl6 33 . . . . . . . . . . . . . . . . . 18  |-  ( E. y ( y  =  s  /\  E. z
( z  =  t  /\  ph ) )  ->  ( y  =  s  ->  ( z  =  t  ->  ph )
) )
4638, 45syl6 33 . . . . . . . . . . . . . . . . 17  |-  ( E. x ( x  =  r  /\  E. y
( y  =  s  /\  E. z ( z  =  t  /\  ph ) ) )  -> 
( x  =  r  ->  ( y  =  s  ->  ( z  =  t  ->  ph )
) ) )
47463impd 1199 . . . . . . . . . . . . . . . 16  |-  ( E. x ( x  =  r  /\  E. y
( y  =  s  /\  E. z ( z  =  t  /\  ph ) ) )  -> 
( ( x  =  r  /\  y  =  s  /\  z  =  t )  ->  ph )
)
4817, 47syl5bi 151 . . . . . . . . . . . . . . 15  |-  ( E. x ( x  =  r  /\  E. y
( y  =  s  /\  E. z ( z  =  t  /\  ph ) ) )  -> 
( <. <. x ,  y
>. ,  z >.  = 
<. <. r ,  s
>. ,  t >.  ->  ph ) )
4948com12 30 . . . . . . . . . . . . . 14  |-  ( <. <. x ,  y >. ,  z >.  =  <. <.
r ,  s >. ,  t >.  ->  ( E. x ( x  =  r  /\  E. y
( y  =  s  /\  E. z ( z  =  t  /\  ph ) ) )  ->  ph ) )
5035, 49syl5 32 . . . . . . . . . . . . 13  |-  ( <. <. x ,  y >. ,  z >.  =  <. <.
r ,  s >. ,  t >.  ->  ( E. x E. y E. z ( <. <. x ,  y >. ,  z
>.  =  <. <. r ,  s >. ,  t
>.  /\  ph )  ->  ph ) )
51 eqeq1 2146 . . . . . . . . . . . . . . 15  |-  ( w  =  <. <. r ,  s
>. ,  t >.  -> 
( w  =  <. <.
x ,  y >. ,  z >.  <->  <. <. r ,  s >. ,  t
>.  =  <. <. x ,  y >. ,  z
>. ) )
52 eqcom 2141 . . . . . . . . . . . . . . 15  |-  ( <. <. r ,  s >. ,  t >.  =  <. <.
x ,  y >. ,  z >.  <->  <. <. x ,  y >. ,  z
>.  =  <. <. r ,  s >. ,  t
>. )
5351, 52syl6bb 195 . . . . . . . . . . . . . 14  |-  ( w  =  <. <. r ,  s
>. ,  t >.  -> 
( w  =  <. <.
x ,  y >. ,  z >.  <->  <. <. x ,  y >. ,  z
>.  =  <. <. r ,  s >. ,  t
>. ) )
5453anbi1d 460 . . . . . . . . . . . . . . . 16  |-  ( w  =  <. <. r ,  s
>. ,  t >.  -> 
( ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  ( <. <.
x ,  y >. ,  z >.  =  <. <.
r ,  s >. ,  t >.  /\  ph ) ) )
55543exbidv 1841 . . . . . . . . . . . . . . 15  |-  ( w  =  <. <. r ,  s
>. ,  t >.  -> 
( E. x E. y E. z ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  E. x E. y E. z (
<. <. x ,  y
>. ,  z >.  = 
<. <. r ,  s
>. ,  t >.  /\ 
ph ) ) )
5655imbi1d 230 . . . . . . . . . . . . . 14  |-  ( w  =  <. <. r ,  s
>. ,  t >.  -> 
( ( E. x E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  ph )  <->  ( E. x E. y E. z (
<. <. x ,  y
>. ,  z >.  = 
<. <. r ,  s
>. ,  t >.  /\ 
ph )  ->  ph )
) )
5753, 56imbi12d 233 . . . . . . . . . . . . 13  |-  ( w  =  <. <. r ,  s
>. ,  t >.  -> 
( ( w  = 
<. <. x ,  y
>. ,  z >.  -> 
( E. x E. y E. z ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  ph )
)  <->  ( <. <. x ,  y >. ,  z
>.  =  <. <. r ,  s >. ,  t
>.  ->  ( E. x E. y E. z (
<. <. x ,  y
>. ,  z >.  = 
<. <. r ,  s
>. ,  t >.  /\ 
ph )  ->  ph )
) ) )
5850, 57mpbiri 167 . . . . . . . . . . . 12  |-  ( w  =  <. <. r ,  s
>. ,  t >.  -> 
( w  =  <. <.
x ,  y >. ,  z >.  ->  ( E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  ph )
) )
5916, 58syl6bi 162 . . . . . . . . . . 11  |-  ( a  =  <. r ,  s
>.  ->  ( w  = 
<. a ,  t >.  ->  ( w  =  <. <.
x ,  y >. ,  z >.  ->  ( E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  ph )
) ) )
6059adantr 274 . . . . . . . . . 10  |-  ( ( a  =  <. r ,  s >.  /\  <. r ,  s >.  =  <. x ,  y >. )  ->  ( w  =  <. a ,  t >.  ->  (
w  =  <. <. x ,  y >. ,  z
>.  ->  ( E. x E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  ph ) ) ) )
6160exlimivv 1868 . . . . . . . . 9  |-  ( E. r E. s ( a  =  <. r ,  s >.  /\  <. r ,  s >.  =  <. x ,  y >. )  ->  ( w  =  <. a ,  t >.  ->  (
w  =  <. <. x ,  y >. ,  z
>.  ->  ( E. x E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  ph ) ) ) )
6214, 61sylbi 120 . . . . . . . 8  |-  ( a  =  <. x ,  y
>.  ->  ( w  = 
<. a ,  t >.  ->  ( w  =  <. <.
x ,  y >. ,  z >.  ->  ( E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  ph )
) ) )
6362com3l 81 . . . . . . 7  |-  ( w  =  <. a ,  t
>.  ->  ( w  = 
<. <. x ,  y
>. ,  z >.  -> 
( a  =  <. x ,  y >.  ->  ( E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  ph )
) ) )
6413, 63mpdd 41 . . . . . 6  |-  ( w  =  <. a ,  t
>.  ->  ( w  = 
<. <. x ,  y
>. ,  z >.  -> 
( E. x E. y E. z ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  ph )
) )
6564adantr 274 . . . . 5  |-  ( ( w  =  <. a ,  t >.  /\  <. a ,  t >.  =  <. <.
x ,  y >. ,  z >. )  ->  ( w  =  <. <.
x ,  y >. ,  z >.  ->  ( E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  ph )
) )
6665exlimivv 1868 . . . 4  |-  ( E. a E. t ( w  =  <. a ,  t >.  /\  <. a ,  t >.  =  <. <.
x ,  y >. ,  z >. )  ->  ( w  =  <. <.
x ,  y >. ,  z >.  ->  ( E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  ph )
) )
678, 66mpcom 36 . . 3  |-  ( w  =  <. <. x ,  y
>. ,  z >.  -> 
( E. x E. y E. z ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  ph )
)
68 19.8a 1569 . . . . 5  |-  ( ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) )
69 19.8a 1569 . . . . 5  |-  ( E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) )
70 19.8a 1569 . . . . 5  |-  ( E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) )
7168, 69, 703syl 17 . . . 4  |-  ( ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) )
7271ex 114 . . 3  |-  ( w  =  <. <. x ,  y
>. ,  z >.  -> 
( ph  ->  E. x E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph ) ) )
7367, 72impbid 128 . 2  |-  ( w  =  <. <. x ,  y
>. ,  z >.  -> 
( E. x E. y E. z ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  ph ) )
74 df-oprab 5778 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }
756, 73, 74elab2 2832 1  |-  ( <. <. x ,  y >. ,  z >.  e.  { <. <. x ,  y
>. ,  z >.  | 
ph }  <->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 962    = wceq 1331   E.wex 1468    e. wcel 1480   E!weu 1999   _Vcvv 2686   <.cop 3530   {coprab 5775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-setind 4452
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-oprab 5778
This theorem is referenced by:  ssoprab2b  5828  ovid  5887  ovidig  5888  tposoprab  6177  xpcomco  6720
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