MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opelopabsb Structured version   Unicode version

Theorem opelopabsb 4730
Description: The law of concretion in terms of substitutions. (Contributed by NM, 30-Sep-2002.) (Revised by Mario Carneiro, 18-Nov-2016.)
Assertion
Ref Expression
opelopabsb  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [. A  /  x ]. [. B  / 
y ]. ph )
Distinct variable groups:    x, y    x, B
Allowed substitution hints:    ph( x, y)    A( x, y)    B( y)

Proof of Theorem opelopabsb
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3083 . . . . . . . . . 10  |-  x  e. 
_V
2 vex 3083 . . . . . . . . . 10  |-  y  e. 
_V
31, 2opnzi 4693 . . . . . . . . 9  |-  <. x ,  y >.  =/=  (/)
4 simpl 458 . . . . . . . . . . 11  |-  ( (
(/)  =  <. x ,  y >.  /\  ph )  ->  (/)  =  <. x ,  y >. )
54eqcomd 2430 . . . . . . . . . 10  |-  ( (
(/)  =  <. x ,  y >.  /\  ph )  ->  <. x ,  y
>.  =  (/) )
65necon3ai 2648 . . . . . . . . 9  |-  ( <.
x ,  y >.  =/=  (/)  ->  -.  ( (/)  =  <. x ,  y
>.  /\  ph ) )
73, 6ax-mp 5 . . . . . . . 8  |-  -.  ( (/)  =  <. x ,  y
>.  /\  ph )
87nex 1672 . . . . . . 7  |-  -.  E. y ( (/)  =  <. x ,  y >.  /\  ph )
98nex 1672 . . . . . 6  |-  -.  E. x E. y ( (/)  =  <. x ,  y
>.  /\  ph )
10 elopab 4728 . . . . . 6  |-  ( (/)  e.  { <. x ,  y
>.  |  ph }  <->  E. x E. y ( (/)  =  <. x ,  y >.  /\  ph ) )
119, 10mtbir 300 . . . . 5  |-  -.  (/)  e.  { <. x ,  y >.  |  ph }
12 eleq1 2495 . . . . 5  |-  ( <. A ,  B >.  =  (/)  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph } 
<->  (/)  e.  { <. x ,  y >.  |  ph } ) )
1311, 12mtbiri 304 . . . 4  |-  ( <. A ,  B >.  =  (/)  ->  -.  <. A ,  B >.  e.  { <. x ,  y >.  |  ph } )
1413necon2ai 2655 . . 3  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  ->  <. A ,  B >.  =/=  (/) )
15 opnz 4692 . . 3  |-  ( <. A ,  B >.  =/=  (/) 
<->  ( A  e.  _V  /\  B  e.  _V )
)
1614, 15sylib 199 . 2  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  ->  ( A  e.  _V  /\  B  e.  _V )
)
17 sbcex 3309 . . 3  |-  ( [. A  /  x ]. [. B  /  y ]. ph  ->  A  e.  _V )
18 spesbc 3381 . . . 4  |-  ( [. A  /  x ]. [. B  /  y ]. ph  ->  E. x [. B  / 
y ]. ph )
19 sbcex 3309 . . . . 5  |-  ( [. B  /  y ]. ph  ->  B  e.  _V )
2019exlimiv 1770 . . . 4  |-  ( E. x [. B  / 
y ]. ph  ->  B  e.  _V )
2118, 20syl 17 . . 3  |-  ( [. A  /  x ]. [. B  /  y ]. ph  ->  B  e.  _V )
2217, 21jca 534 . 2  |-  ( [. A  /  x ]. [. B  /  y ]. ph  ->  ( A  e.  _V  /\  B  e.  _V )
)
23 opeq1 4187 . . . . 5  |-  ( z  =  A  ->  <. z ,  w >.  =  <. A ,  w >. )
2423eleq1d 2491 . . . 4  |-  ( z  =  A  ->  ( <. z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  <. A ,  w >.  e.  { <. x ,  y >.  |  ph } ) )
25 dfsbcq2 3302 . . . 4  |-  ( z  =  A  ->  ( [ z  /  x ] [ w  /  y ] ph  <->  [. A  /  x ]. [ w  /  y ] ph ) )
2624, 25bibi12d 322 . . 3  |-  ( z  =  A  ->  (
( <. z ,  w >.  e.  { <. x ,  y >.  |  ph } 
<->  [ z  /  x ] [ w  /  y ] ph )  <->  ( <. A ,  w >.  e.  { <. x ,  y >.  |  ph }  <->  [. A  /  x ]. [ w  / 
y ] ph )
) )
27 opeq2 4188 . . . . 5  |-  ( w  =  B  ->  <. A ,  w >.  =  <. A ,  B >. )
2827eleq1d 2491 . . . 4  |-  ( w  =  B  ->  ( <. A ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ph } ) )
29 dfsbcq2 3302 . . . . 5  |-  ( w  =  B  ->  ( [ w  /  y ] ph  <->  [. B  /  y ]. ph ) )
3029sbcbidv 3354 . . . 4  |-  ( w  =  B  ->  ( [. A  /  x ]. [ w  /  y ] ph  <->  [. A  /  x ]. [. B  /  y ]. ph ) )
3128, 30bibi12d 322 . . 3  |-  ( w  =  B  ->  (
( <. A ,  w >.  e.  { <. x ,  y >.  |  ph } 
<-> 
[. A  /  x ]. [ w  /  y ] ph )  <->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph }  <->  [. A  /  x ]. [. B  / 
y ]. ph ) ) )
32 nfopab1 4490 . . . . . 6  |-  F/_ x { <. x ,  y
>.  |  ph }
3332nfel2 2598 . . . . 5  |-  F/ x <. z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }
34 nfs1v 2236 . . . . 5  |-  F/ x [ z  /  x ] [ w  /  y ] ph
3533, 34nfbi 1994 . . . 4  |-  F/ x
( <. z ,  w >.  e.  { <. x ,  y >.  |  ph } 
<->  [ z  /  x ] [ w  /  y ] ph )
36 opeq1 4187 . . . . . 6  |-  ( x  =  z  ->  <. x ,  w >.  =  <. z ,  w >. )
3736eleq1d 2491 . . . . 5  |-  ( x  =  z  ->  ( <. x ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  <. z ,  w >.  e.  { <. x ,  y >.  |  ph } ) )
38 sbequ12 2051 . . . . 5  |-  ( x  =  z  ->  ( [ w  /  y ] ph  <->  [ z  /  x ] [ w  /  y ] ph ) )
3937, 38bibi12d 322 . . . 4  |-  ( x  =  z  ->  (
( <. x ,  w >.  e.  { <. x ,  y >.  |  ph } 
<->  [ w  /  y ] ph )  <->  ( <. z ,  w >.  e.  { <. x ,  y >.  |  ph }  <->  [ z  /  x ] [ w  /  y ] ph ) ) )
40 nfopab2 4491 . . . . . . 7  |-  F/_ y { <. x ,  y
>.  |  ph }
4140nfel2 2598 . . . . . 6  |-  F/ y
<. x ,  w >.  e. 
{ <. x ,  y
>.  |  ph }
42 nfs1v 2236 . . . . . 6  |-  F/ y [ w  /  y ] ph
4341, 42nfbi 1994 . . . . 5  |-  F/ y ( <. x ,  w >.  e.  { <. x ,  y >.  |  ph } 
<->  [ w  /  y ] ph )
44 opeq2 4188 . . . . . . 7  |-  ( y  =  w  ->  <. x ,  y >.  =  <. x ,  w >. )
4544eleq1d 2491 . . . . . 6  |-  ( y  =  w  ->  ( <. x ,  y >.  e.  { <. x ,  y
>.  |  ph }  <->  <. x ,  w >.  e.  { <. x ,  y >.  |  ph } ) )
46 sbequ12 2051 . . . . . 6  |-  ( y  =  w  ->  ( ph 
<->  [ w  /  y ] ph ) )
4745, 46bibi12d 322 . . . . 5  |-  ( y  =  w  ->  (
( <. x ,  y
>.  e.  { <. x ,  y >.  |  ph } 
<-> 
ph )  <->  ( <. x ,  w >.  e.  { <. x ,  y >.  |  ph }  <->  [ w  /  y ] ph ) ) )
48 opabid 4727 . . . . 5  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ph }  <->  ph )
4943, 47, 48chvar 2071 . . . 4  |-  ( <.
x ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [ w  /  y ] ph )
5035, 39, 49chvar 2071 . . 3  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [ z  /  x ] [ w  /  y ] ph )
5126, 31, 50vtocl2g 3143 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph } 
<-> 
[. A  /  x ]. [. B  /  y ]. ph ) )
5216, 22, 51pm5.21nii 354 1  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [. A  /  x ]. [. B  / 
y ]. ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1657   [wsb 1790    e. wcel 1872    =/= wne 2614   _Vcvv 3080   [.wsbc 3299   (/)c0 3761   <.cop 4004   {copab 4481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-opab 4483
This theorem is referenced by:  brabsb  4731  opelopabgf  4740  opelopabaf  4744  opelopabf  4745  difopab  4985  isarep1  5680  fmptsnd  6101
  Copyright terms: Public domain W3C validator