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Theorem cnvoprab 5883
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 1570 . . . . . 6  |-  ( E. a E. z ( w  =  <. a ,  z >.  /\  ps ) 
<->  E. z E. a
( w  =  <. a ,  z >.  /\  ps ) )
2 nfv 1437 . . . . . . . . . . 11  |-  F/ x  w  =  <. a ,  z >.
3 cnvoprab.x . . . . . . . . . . 11  |-  F/ x ps
42, 3nfan 1473 . . . . . . . . . 10  |-  F/ x
( w  =  <. a ,  z >.  /\  ps )
54nfex 1544 . . . . . . . . 9  |-  F/ x E. a ( w  = 
<. a ,  z >.  /\  ps )
6 nfv 1437 . . . . . . . . . . . 12  |-  F/ y  w  =  <. a ,  z >.
7 cnvoprab.y . . . . . . . . . . . 12  |-  F/ y ps
86, 7nfan 1473 . . . . . . . . . . 11  |-  F/ y ( w  =  <. a ,  z >.  /\  ps )
98nfex 1544 . . . . . . . . . 10  |-  F/ y E. a ( w  =  <. a ,  z
>.  /\  ps )
10 vex 2577 . . . . . . . . . . . 12  |-  x  e. 
_V
11 vex 2577 . . . . . . . . . . . 12  |-  y  e. 
_V
1210, 11opex 3994 . . . . . . . . . . 11  |-  <. x ,  y >.  e.  _V
13 opeq1 3577 . . . . . . . . . . . . 13  |-  ( a  =  <. x ,  y
>.  ->  <. a ,  z
>.  =  <. <. x ,  y >. ,  z
>. )
1413eqeq2d 2067 . . . . . . . . . . . 12  |-  ( a  =  <. x ,  y
>.  ->  ( w  = 
<. a ,  z >.  <->  w  =  <. <. x ,  y
>. ,  z >. ) )
15 cnvoprab.1 . . . . . . . . . . . 12  |-  ( a  =  <. x ,  y
>.  ->  ( ps  <->  ph ) )
1614, 15anbi12d 450 . . . . . . . . . . 11  |-  ( a  =  <. x ,  y
>.  ->  ( ( w  =  <. a ,  z
>.  /\  ps )  <->  ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ph ) ) )
1712, 16spcev 2664 . . . . . . . . . 10  |-  ( ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  E. a ( w  = 
<. a ,  z >.  /\  ps ) )
189, 17exlimi 1501 . . . . . . . . 9  |-  ( E. y ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  E. a
( w  =  <. a ,  z >.  /\  ps ) )
195, 18exlimi 1501 . . . . . . . 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 266 . . . . . . . . . 10  |-  ( ( w  =  <. a ,  z >.  /\  ps )  ->  a  e.  ( _V  X.  _V )
)
22 vex 2577 . . . . . . . . . . . 12  |-  a  e. 
_V
23 1stexg 5822 . . . . . . . . . . . 12  |-  ( a  e.  _V  ->  ( 1st `  a )  e. 
_V )
2422, 23ax-mp 7 . . . . . . . . . . 11  |-  ( 1st `  a )  e.  _V
25 2ndexg 5823 . . . . . . . . . . . 12  |-  ( a  e.  _V  ->  ( 2nd `  a )  e. 
_V )
2622, 25ax-mp 7 . . . . . . . . . . 11  |-  ( 2nd `  a )  e.  _V
27 eqcom 2058 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  a )  =  x  <->  x  =  ( 1st `  a ) )
28 eqcom 2058 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  a )  =  y  <->  y  =  ( 2nd `  a ) )
2927, 28anbi12i 441 . . . . . . . . . . . . . 14  |-  ( ( ( 1st `  a
)  =  x  /\  ( 2nd `  a )  =  y )  <->  ( x  =  ( 1st `  a
)  /\  y  =  ( 2nd `  a ) ) )
30 eqopi 5826 . . . . . . . . . . . . . 14  |-  ( ( a  e.  ( _V 
X.  _V )  /\  (
( 1st `  a
)  =  x  /\  ( 2nd `  a )  =  y ) )  ->  a  =  <. x ,  y >. )
3129, 30sylan2br 276 . . . . . . . . . . . . 13  |-  ( ( a  e.  ( _V 
X.  _V )  /\  (
x  =  ( 1st `  a )  /\  y  =  ( 2nd `  a
) ) )  -> 
a  =  <. x ,  y >. )
3216bicomd 133 . . . . . . . . . . . . 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 5882 . . . . . . . . . . 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 423 . . . . . . . . . 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 1505 . . . . . . . 8  |-  ( E. a ( w  = 
<. a ,  z >.  /\  ps )  ->  E. x E. y ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) )
3819, 37impbii 121 . . . . . . 7  |-  ( E. x E. y ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph )  <->  E. a
( w  =  <. a ,  z >.  /\  ps ) )
3938exbii 1512 . . . . . 6  |-  ( E. z E. x E. y ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  E. z E. a ( w  = 
<. a ,  z >.  /\  ps ) )
40 exrot3 1596 . . . . . 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 202 . . . . 5  |-  ( E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  E. a E. z ( w  = 
<. a ,  z >.  /\  ps ) )
4241abbii 2169 . . . 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 5544 . . . 4  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }
44 df-opab 3847 . . . 4  |-  { <. a ,  z >.  |  ps }  =  { w  |  E. a E. z
( w  =  <. a ,  z >.  /\  ps ) }
4542, 43, 443eqtr4ri 2087 . . 3  |-  { <. a ,  z >.  |  ps }  =  { <. <. x ,  y >. ,  z
>.  |  ph }
4645cnveqi 4538 . 2  |-  `' { <. a ,  z >.  |  ps }  =  `' { <. <. x ,  y
>. ,  z >.  | 
ph }
47 cnvopab 4754 . 2  |-  `' { <. a ,  z >.  |  ps }  =  { <. z ,  a >.  |  ps }
4846, 47eqtr3i 2078 1  |-  `' { <. <. x ,  y
>. ,  z >.  | 
ph }  =  { <. z ,  a >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259   F/wnf 1365   E.wex 1397    e. wcel 1409   {cab 2042   _Vcvv 2574   <.cop 3406   {copab 3845    X. cxp 4371   `'ccnv 4372   ` cfv 4930   {coprab 5541   1stc1st 5793   2ndc2nd 5794
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-fo 4936  df-fv 4938  df-oprab 5544  df-1st 5795  df-2nd 5796
This theorem is referenced by:  f1od2  5884
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