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Theorem cnvoprab 5981
Description: The converse of a class abstraction of nested ordered pairs. (Contributed by Thierry Arnoux, 17-Aug-2017.)
Hypotheses
Ref Expression
cnvoprab.x  |-  F/ x ps
cnvoprab.y  |-  F/ y ps
cnvoprab.1  |-  ( a  =  <. x ,  y
>.  ->  ( ps  <->  ph ) )
cnvoprab.2  |-  ( ps 
->  a  e.  ( _V  X.  _V ) )
Assertion
Ref Expression
cnvoprab  |-  `' { <. <. x ,  y
>. ,  z >.  | 
ph }  =  { <. z ,  a >.  |  ps }
Distinct variable groups:    x, a, y, z    ph, a
Allowed substitution hints:    ph( x, y, z)    ps( x, y, z, a)

Proof of Theorem cnvoprab
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 excom 1599 . . . . . 6  |-  ( E. a E. z ( w  =  <. a ,  z >.  /\  ps ) 
<->  E. z E. a
( w  =  <. a ,  z >.  /\  ps ) )
2 nfv 1466 . . . . . . . . . . 11  |-  F/ x  w  =  <. a ,  z >.
3 cnvoprab.x . . . . . . . . . . 11  |-  F/ x ps
42, 3nfan 1502 . . . . . . . . . 10  |-  F/ x
( w  =  <. a ,  z >.  /\  ps )
54nfex 1573 . . . . . . . . 9  |-  F/ x E. a ( w  = 
<. a ,  z >.  /\  ps )
6 nfv 1466 . . . . . . . . . . . 12  |-  F/ y  w  =  <. a ,  z >.
7 cnvoprab.y . . . . . . . . . . . 12  |-  F/ y ps
86, 7nfan 1502 . . . . . . . . . . 11  |-  F/ y ( w  =  <. a ,  z >.  /\  ps )
98nfex 1573 . . . . . . . . . 10  |-  F/ y E. a ( w  =  <. a ,  z
>.  /\  ps )
10 vex 2622 . . . . . . . . . . . 12  |-  x  e. 
_V
11 vex 2622 . . . . . . . . . . . 12  |-  y  e. 
_V
1210, 11opex 4047 . . . . . . . . . . 11  |-  <. x ,  y >.  e.  _V
13 opeq1 3617 . . . . . . . . . . . . 13  |-  ( a  =  <. x ,  y
>.  ->  <. a ,  z
>.  =  <. <. x ,  y >. ,  z
>. )
1413eqeq2d 2099 . . . . . . . . . . . 12  |-  ( a  =  <. x ,  y
>.  ->  ( w  = 
<. a ,  z >.  <->  w  =  <. <. x ,  y
>. ,  z >. ) )
15 cnvoprab.1 . . . . . . . . . . . 12  |-  ( a  =  <. x ,  y
>.  ->  ( ps  <->  ph ) )
1614, 15anbi12d 457 . . . . . . . . . . 11  |-  ( a  =  <. x ,  y
>.  ->  ( ( w  =  <. a ,  z
>.  /\  ps )  <->  ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ph ) ) )
1712, 16spcev 2713 . . . . . . . . . 10  |-  ( ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  E. a ( w  = 
<. a ,  z >.  /\  ps ) )
189, 17exlimi 1530 . . . . . . . . 9  |-  ( E. y ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  E. a
( w  =  <. a ,  z >.  /\  ps ) )
195, 18exlimi 1530 . . . . . . . 8  |-  ( E. x E. y ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  E. a ( w  = 
<. a ,  z >.  /\  ps ) )
20 cnvoprab.2 . . . . . . . . . . 11  |-  ( ps 
->  a  e.  ( _V  X.  _V ) )
2120adantl 271 . . . . . . . . . 10  |-  ( ( w  =  <. a ,  z >.  /\  ps )  ->  a  e.  ( _V  X.  _V )
)
22 vex 2622 . . . . . . . . . . . 12  |-  a  e. 
_V
23 1stexg 5920 . . . . . . . . . . . 12  |-  ( a  e.  _V  ->  ( 1st `  a )  e. 
_V )
2422, 23ax-mp 7 . . . . . . . . . . 11  |-  ( 1st `  a )  e.  _V
25 2ndexg 5921 . . . . . . . . . . . 12  |-  ( a  e.  _V  ->  ( 2nd `  a )  e. 
_V )
2622, 25ax-mp 7 . . . . . . . . . . 11  |-  ( 2nd `  a )  e.  _V
27 eqcom 2090 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  a )  =  x  <->  x  =  ( 1st `  a ) )
28 eqcom 2090 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  a )  =  y  <->  y  =  ( 2nd `  a ) )
2927, 28anbi12i 448 . . . . . . . . . . . . . 14  |-  ( ( ( 1st `  a
)  =  x  /\  ( 2nd `  a )  =  y )  <->  ( x  =  ( 1st `  a
)  /\  y  =  ( 2nd `  a ) ) )
30 eqopi 5924 . . . . . . . . . . . . . 14  |-  ( ( a  e.  ( _V 
X.  _V )  /\  (
( 1st `  a
)  =  x  /\  ( 2nd `  a )  =  y ) )  ->  a  =  <. x ,  y >. )
3129, 30sylan2br 282 . . . . . . . . . . . . 13  |-  ( ( a  e.  ( _V 
X.  _V )  /\  (
x  =  ( 1st `  a )  /\  y  =  ( 2nd `  a
) ) )  -> 
a  =  <. x ,  y >. )
3216bicomd 139 . . . . . . . . . . . . 13  |-  ( a  =  <. x ,  y
>.  ->  ( ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  ( w  =  <. a ,  z
>.  /\  ps ) ) )
3331, 32syl 14 . . . . . . . . . . . 12  |-  ( ( a  e.  ( _V 
X.  _V )  /\  (
x  =  ( 1st `  a )  /\  y  =  ( 2nd `  a
) ) )  -> 
( ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  ( w  =  <. a ,  z
>.  /\  ps ) ) )
344, 8, 33spc2ed 5980 . . . . . . . . . . 11  |-  ( ( a  e.  ( _V 
X.  _V )  /\  (
( 1st `  a
)  e.  _V  /\  ( 2nd `  a )  e.  _V ) )  ->  ( ( w  =  <. a ,  z
>.  /\  ps )  ->  E. x E. y ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph ) ) )
3524, 26, 34mpanr12 430 . . . . . . . . . 10  |-  ( a  e.  ( _V  X.  _V )  ->  ( ( w  =  <. a ,  z >.  /\  ps )  ->  E. x E. y
( w  =  <. <.
x ,  y >. ,  z >.  /\  ph ) ) )
3621, 35mpcom 36 . . . . . . . . 9  |-  ( ( w  =  <. a ,  z >.  /\  ps )  ->  E. x E. y
( w  =  <. <.
x ,  y >. ,  z >.  /\  ph ) )
3736exlimiv 1534 . . . . . . . 8  |-  ( E. a ( w  = 
<. a ,  z >.  /\  ps )  ->  E. x E. y ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) )
3819, 37impbii 124 . . . . . . 7  |-  ( E. x E. y ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph )  <->  E. a
( w  =  <. a ,  z >.  /\  ps ) )
3938exbii 1541 . . . . . 6  |-  ( E. z E. x E. y ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  E. z E. a ( w  = 
<. a ,  z >.  /\  ps ) )
40 exrot3 1625 . . . . . 6  |-  ( E. z E. x E. y ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  E. x E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph ) )
411, 39, 403bitr2ri 207 . . . . 5  |-  ( E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  E. a E. z ( w  = 
<. a ,  z >.  /\  ps ) )
4241abbii 2203 . . . 4  |-  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }  =  { w  |  E. a E. z ( w  =  <. a ,  z
>.  /\  ps ) }
43 df-oprab 5638 . . . 4  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }
44 df-opab 3892 . . . 4  |-  { <. a ,  z >.  |  ps }  =  { w  |  E. a E. z
( w  =  <. a ,  z >.  /\  ps ) }
4542, 43, 443eqtr4ri 2119 . . 3  |-  { <. a ,  z >.  |  ps }  =  { <. <. x ,  y >. ,  z
>.  |  ph }
4645cnveqi 4599 . 2  |-  `' { <. a ,  z >.  |  ps }  =  `' { <. <. x ,  y
>. ,  z >.  | 
ph }
47 cnvopab 4820 . 2  |-  `' { <. a ,  z >.  |  ps }  =  { <. z ,  a >.  |  ps }
4846, 47eqtr3i 2110 1  |-  `' { <. <. x ,  y
>. ,  z >.  | 
ph }  =  { <. z ,  a >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289   F/wnf 1394   E.wex 1426    e. wcel 1438   {cab 2074   _Vcvv 2619   <.cop 3444   {copab 3890    X. cxp 4426   `'ccnv 4427   ` cfv 5002   {coprab 5635   1stc1st 5891   2ndc2nd 5892
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fo 5008  df-fv 5010  df-oprab 5638  df-1st 5893  df-2nd 5894
This theorem is referenced by:  f1od2  5982
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