ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eloprabga Unicode version

Theorem eloprabga 5622
Description: The law of concretion for operation class abstraction. Compare elopab 4021. (Contributed by NM, 14-Sep-1999.) (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 19-Dec-2013.)
Hypothesis
Ref Expression
eloprabga.1  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
eloprabga  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( <. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y
>. ,  z >.  | 
ph }  <->  ps )
)
Distinct variable groups:    x, y, z, A    x, B, y, z    x, C, y, z    ps, x, y, z
Allowed substitution hints:    ph( x, y, z)    V( x, y, z)    W( x, y, z)    X( x, y, z)

Proof of Theorem eloprabga
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 elex 2611 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 elex 2611 . 2  |-  ( B  e.  W  ->  B  e.  _V )
3 elex 2611 . 2  |-  ( C  e.  X  ->  C  e.  _V )
4 opexg 3991 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  _V )  -> 
<. A ,  B >.  e. 
_V )
5 opexg 3991 . . . . 5  |-  ( (
<. A ,  B >.  e. 
_V  /\  C  e.  _V )  ->  <. <. A ,  B >. ,  C >.  e. 
_V )
64, 5sylan 277 . . . 4  |-  ( ( ( A  e.  _V  /\  B  e.  _V )  /\  C  e.  _V )  ->  <. <. A ,  B >. ,  C >.  e.  _V )
763impa 1134 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  <. <. A ,  B >. ,  C >.  e. 
_V )
8 simpr 108 . . . . . . . . . . 11  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  w  =  <. <. A ,  B >. ,  C >. )  ->  w  =  <. <. A ,  B >. ,  C >. )
98eqeq1d 2090 . . . . . . . . . 10  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  w  =  <. <. A ,  B >. ,  C >. )  ->  (
w  =  <. <. x ,  y >. ,  z
>. 
<-> 
<. <. A ,  B >. ,  C >.  =  <. <.
x ,  y >. ,  z >. )
)
10 eqcom 2084 . . . . . . . . . . 11  |-  ( <. <. A ,  B >. ,  C >.  =  <. <.
x ,  y >. ,  z >.  <->  <. <. x ,  y >. ,  z
>.  =  <. <. A ,  B >. ,  C >. )
11 vex 2605 . . . . . . . . . . . 12  |-  x  e. 
_V
12 vex 2605 . . . . . . . . . . . 12  |-  y  e. 
_V
13 vex 2605 . . . . . . . . . . . 12  |-  z  e. 
_V
1411, 12, 13otth2 4004 . . . . . . . . . . 11  |-  ( <. <. x ,  y >. ,  z >.  =  <. <. A ,  B >. ,  C >.  <->  ( x  =  A  /\  y  =  B  /\  z  =  C ) )
1510, 14bitri 182 . . . . . . . . . 10  |-  ( <. <. A ,  B >. ,  C >.  =  <. <.
x ,  y >. ,  z >.  <->  ( x  =  A  /\  y  =  B  /\  z  =  C ) )
169, 15syl6bb 194 . . . . . . . . 9  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  w  =  <. <. A ,  B >. ,  C >. )  ->  (
w  =  <. <. x ,  y >. ,  z
>. 
<->  ( x  =  A  /\  y  =  B  /\  z  =  C ) ) )
1716anbi1d 453 . . . . . . . 8  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  w  =  <. <. A ,  B >. ,  C >. )  ->  (
( w  =  <. <.
x ,  y >. ,  z >.  /\  ph ) 
<->  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ph ) ) )
18 eloprabga.1 . . . . . . . . 9  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ph  <->  ps )
)
1918pm5.32i 442 . . . . . . . 8  |-  ( ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ph )  <->  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ps )
)
2017, 19syl6bb 194 . . . . . . 7  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  w  =  <. <. A ,  B >. ,  C >. )  ->  (
( w  =  <. <.
x ,  y >. ,  z >.  /\  ph ) 
<->  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ps ) ) )
21203exbidv 1791 . . . . . 6  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  w  =  <. <. A ,  B >. ,  C >. )  ->  ( E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  E. x E. y E. z ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ps )
) )
22 df-oprab 5547 . . . . . . . . . 10  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }
2322eleq2i 2146 . . . . . . . . 9  |-  ( w  e.  { <. <. x ,  y >. ,  z
>.  |  ph }  <->  w  e.  { w  |  E. x E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph ) } )
24 abid 2070 . . . . . . . . 9  |-  ( w  e.  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }  <->  E. x E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph ) )
2523, 24bitr2i 183 . . . . . . . 8  |-  ( E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  w  e.  {
<. <. x ,  y
>. ,  z >.  | 
ph } )
26 eleq1 2142 . . . . . . . 8  |-  ( w  =  <. <. A ,  B >. ,  C >.  ->  (
w  e.  { <. <.
x ,  y >. ,  z >.  |  ph } 
<-> 
<. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y
>. ,  z >.  | 
ph } ) )
2725, 26syl5bb 190 . . . . . . 7  |-  ( w  =  <. <. A ,  B >. ,  C >.  ->  ( E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  <. <. A ,  B >. ,  C >.  e. 
{ <. <. x ,  y
>. ,  z >.  | 
ph } ) )
2827adantl 271 . . . . . 6  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  w  =  <. <. A ,  B >. ,  C >. )  ->  ( E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  <. <. A ,  B >. ,  C >.  e. 
{ <. <. x ,  y
>. ,  z >.  | 
ph } ) )
29 elisset 2614 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  E. x  x  =  A )
30 elisset 2614 . . . . . . . . . . 11  |-  ( B  e.  _V  ->  E. y 
y  =  B )
31 elisset 2614 . . . . . . . . . . 11  |-  ( C  e.  _V  ->  E. z 
z  =  C )
3229, 30, 313anim123i 1124 . . . . . . . . . 10  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( E. x  x  =  A  /\  E. y  y  =  B  /\  E. z  z  =  C
) )
33 eeeanv 1850 . . . . . . . . . 10  |-  ( E. x E. y E. z ( x  =  A  /\  y  =  B  /\  z  =  C )  <->  ( E. x  x  =  A  /\  E. y  y  =  B  /\  E. z 
z  =  C ) )
3432, 33sylibr 132 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  E. x E. y E. z ( x  =  A  /\  y  =  B  /\  z  =  C )
)
3534biantrurd 299 . . . . . . . 8  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( ps 
<->  ( E. x E. y E. z ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ps ) ) )
36 19.41vvv 1826 . . . . . . . 8  |-  ( E. x E. y E. z ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ps ) 
<->  ( E. x E. y E. z ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ps ) )
3735, 36syl6rbbr 197 . . . . . . 7  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( E. x E. y E. z ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ps ) 
<->  ps ) )
3837adantr 270 . . . . . 6  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  w  =  <. <. A ,  B >. ,  C >. )  ->  ( E. x E. y E. z ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ps ) 
<->  ps ) )
3921, 28, 383bitr3d 216 . . . . 5  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  w  =  <. <. A ,  B >. ,  C >. )  ->  ( <. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y
>. ,  z >.  | 
ph }  <->  ps )
)
4039expcom 114 . . . 4  |-  ( w  =  <. <. A ,  B >. ,  C >.  ->  (
( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( <. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y
>. ,  z >.  | 
ph }  <->  ps )
) )
4140vtocleg 2670 . . 3  |-  ( <. <. A ,  B >. ,  C >.  e.  _V  ->  ( ( A  e. 
_V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( <. <. A ,  B >. ,  C >.  e. 
{ <. <. x ,  y
>. ,  z >.  | 
ph }  <->  ps )
) )
427, 41mpcom 36 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( <. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y
>. ,  z >.  | 
ph }  <->  ps )
)
431, 2, 3, 42syl3an 1212 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( <. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y
>. ,  z >.  | 
ph }  <->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 920    = wceq 1285   E.wex 1422    e. wcel 1434   {cab 2068   _Vcvv 2602   <.cop 3409   {coprab 5544
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-oprab 5547
This theorem is referenced by:  eloprabg  5623  ovigg  5652
  Copyright terms: Public domain W3C validator