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Theorem cnvoprab 26045
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 1787 . . . . . 6  |-  ( E. a E. z ( w  =  <. a ,  z >.  /\  ps ) 
<->  E. z E. a
( w  =  <. a ,  z >.  /\  ps ) )
2 opex 4577 . . . . . . . . . . . . 13  |-  <. x ,  y >.  e.  _V
3 opeq1 4080 . . . . . . . . . . . . . . . 16  |-  ( a  =  <. x ,  y
>.  ->  <. a ,  z
>.  =  <. <. x ,  y >. ,  z
>. )
43eqeq2d 2454 . . . . . . . . . . . . . . 15  |-  ( a  =  <. x ,  y
>.  ->  ( w  = 
<. a ,  z >.  <->  w  =  <. <. x ,  y
>. ,  z >. ) )
5 cnvoprab.1 . . . . . . . . . . . . . . 15  |-  ( a  =  <. x ,  y
>.  ->  ( ps  <->  ph ) )
64, 5anbi12d 710 . . . . . . . . . . . . . 14  |-  ( a  =  <. x ,  y
>.  ->  ( ( w  =  <. a ,  z
>.  /\  ps )  <->  ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ph ) ) )
76spcegv 3079 . . . . . . . . . . . . 13  |-  ( <.
x ,  y >.  e.  _V  ->  ( (
w  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  E. a ( w  = 
<. a ,  z >.  /\  ps ) ) )
82, 7ax-mp 5 . . . . . . . . . . . 12  |-  ( ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  E. a ( w  = 
<. a ,  z >.  /\  ps ) )
98eximi 1625 . . . . . . . . . . 11  |-  ( E. y ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  E. y E. a ( w  = 
<. a ,  z >.  /\  ps ) )
10 nfv 1673 . . . . . . . . . . . . . 14  |-  F/ y  w  =  <. a ,  z >.
11 cnvoprab.y . . . . . . . . . . . . . 14  |-  F/ y ps
1210, 11nfan 1861 . . . . . . . . . . . . 13  |-  F/ y ( w  =  <. a ,  z >.  /\  ps )
1312nfex 1874 . . . . . . . . . . . 12  |-  F/ y E. a ( w  =  <. a ,  z
>.  /\  ps )
141319.9 1827 . . . . . . . . . . 11  |-  ( E. y E. a ( w  =  <. a ,  z >.  /\  ps ) 
<->  E. a ( w  =  <. a ,  z
>.  /\  ps ) )
159, 14sylib 196 . . . . . . . . . 10  |-  ( E. y ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  E. a
( w  =  <. a ,  z >.  /\  ps ) )
1615eximi 1625 . . . . . . . . 9  |-  ( E. x E. y ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  E. x E. a ( w  =  <. a ,  z >.  /\  ps ) )
17 nfv 1673 . . . . . . . . . . . 12  |-  F/ x  w  =  <. a ,  z >.
18 cnvoprab.x . . . . . . . . . . . 12  |-  F/ x ps
1917, 18nfan 1861 . . . . . . . . . . 11  |-  F/ x
( w  =  <. a ,  z >.  /\  ps )
2019nfex 1874 . . . . . . . . . 10  |-  F/ x E. a ( w  = 
<. a ,  z >.  /\  ps )
212019.9 1827 . . . . . . . . 9  |-  ( E. x E. a ( w  =  <. a ,  z >.  /\  ps ) 
<->  E. a ( w  =  <. a ,  z
>.  /\  ps ) )
2216, 21sylib 196 . . . . . . . 8  |-  ( E. x E. y ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  E. a ( w  = 
<. a ,  z >.  /\  ps ) )
23 cnvoprab.2 . . . . . . . . . . 11  |-  ( ps 
->  a  e.  ( _V  X.  _V ) )
2423adantl 466 . . . . . . . . . 10  |-  ( ( w  =  <. a ,  z >.  /\  ps )  ->  a  e.  ( _V  X.  _V )
)
25 fvex 5722 . . . . . . . . . . 11  |-  ( 1st `  a )  e.  _V
26 fvex 5722 . . . . . . . . . . 11  |-  ( 2nd `  a )  e.  _V
27 eqcom 2445 . . . . . . . . . . . . . . . 16  |-  ( ( 1st `  a )  =  x  <->  x  =  ( 1st `  a ) )
28 eqcom 2445 . . . . . . . . . . . . . . . 16  |-  ( ( 2nd `  a )  =  y  <->  y  =  ( 2nd `  a ) )
2927, 28anbi12i 697 . . . . . . . . . . . . . . 15  |-  ( ( ( 1st `  a
)  =  x  /\  ( 2nd `  a )  =  y )  <->  ( x  =  ( 1st `  a
)  /\  y  =  ( 2nd `  a ) ) )
30 eqopi 6631 . . . . . . . . . . . . . . 15  |-  ( ( a  e.  ( _V 
X.  _V )  /\  (
( 1st `  a
)  =  x  /\  ( 2nd `  a )  =  y ) )  ->  a  =  <. x ,  y >. )
3129, 30sylan2br 476 . . . . . . . . . . . . . 14  |-  ( ( a  e.  ( _V 
X.  _V )  /\  (
x  =  ( 1st `  a )  /\  y  =  ( 2nd `  a
) ) )  -> 
a  =  <. x ,  y >. )
326bicomd 201 . . . . . . . . . . . . . 14  |-  ( a  =  <. x ,  y
>.  ->  ( ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  ( w  =  <. a ,  z
>.  /\  ps ) ) )
3331, 32syl 16 . . . . . . . . . . . . 13  |-  ( ( a  e.  ( _V 
X.  _V )  /\  (
x  =  ( 1st `  a )  /\  y  =  ( 2nd `  a
) ) )  -> 
( ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  ( w  =  <. a ,  z
>.  /\  ps ) ) )
3419, 12, 33spc2ed 25879 . . . . . . . . . . . 12  |-  ( ( 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 ) ) )
3534ex 434 . . . . . . . . . . 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 ) ) ) )
3625, 26, 35mp2ani 678 . . . . . . . . . 10  |-  ( a  e.  ( _V  X.  _V )  ->  ( ( w  =  <. a ,  z >.  /\  ps )  ->  E. x E. y
( w  =  <. <.
x ,  y >. ,  z >.  /\  ph ) ) )
3724, 36mpcom 36 . . . . . . . . 9  |-  ( ( w  =  <. a ,  z >.  /\  ps )  ->  E. x E. y
( w  =  <. <.
x ,  y >. ,  z >.  /\  ph ) )
3837exlimiv 1688 . . . . . . . 8  |-  ( E. a ( w  = 
<. a ,  z >.  /\  ps )  ->  E. x E. y ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) )
3922, 38impbii 188 . . . . . . 7  |-  ( E. x E. y ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph )  <->  E. a
( w  =  <. a ,  z >.  /\  ps ) )
4039exbii 1634 . . . . . 6  |-  ( E. z E. x E. y ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  E. z E. a ( w  = 
<. a ,  z >.  /\  ps ) )
41 excom 1787 . . . . . . . 8  |-  ( E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph )  <->  E. z E. y ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) )
4241exbii 1634 . . . . . . 7  |-  ( E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  E. x E. z E. y ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph ) )
43 excom 1787 . . . . . . 7  |-  ( E. x E. z E. y ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  E. z E. x E. y ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph ) )
4442, 43bitr2i 250 . . . . . 6  |-  ( E. z E. x E. y ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  E. x E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph ) )
451, 40, 443bitr2ri 274 . . . . 5  |-  ( E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  E. a E. z ( w  = 
<. a ,  z >.  /\  ps ) )
4645abbii 2561 . . . 4  |-  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }  =  { w  |  E. a E. z ( w  =  <. a ,  z
>.  /\  ps ) }
47 df-oprab 6116 . . . 4  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }
48 df-opab 4372 . . . 4  |-  { <. a ,  z >.  |  ps }  =  { w  |  E. a E. z
( w  =  <. a ,  z >.  /\  ps ) }
4946, 47, 483eqtr4ri 2474 . . 3  |-  { <. a ,  z >.  |  ps }  =  { <. <. x ,  y >. ,  z
>.  |  ph }
5049cnveqi 5035 . 2  |-  `' { <. a ,  z >.  |  ps }  =  `' { <. <. x ,  y
>. ,  z >.  | 
ph }
51 cnvopab 5259 . 2  |-  `' { <. a ,  z >.  |  ps }  =  { <. z ,  a >.  |  ps }
5250, 51eqtr3i 2465 1  |-  `' { <. <. x ,  y
>. ,  z >.  | 
ph }  =  { <. z ,  a >.  |  ps }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   E.wex 1586   F/wnf 1589    e. wcel 1756   {cab 2429   _Vcvv 2993   <.cop 3904   {copab 4370    X. cxp 4859   `'ccnv 4860   ` cfv 5439   {coprab 6113   1stc1st 6596   2ndc2nd 6597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-iota 5402  df-fun 5441  df-fv 5447  df-oprab 6116  df-1st 6598  df-2nd 6599
This theorem is referenced by:  f1od2  26046
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