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Theorem bj-elid 31384
Description: Characterization of the elements of  _I. (Contributed by BJ, 22-Jun-2019.)
Assertion
Ref Expression
bj-elid  |-  ( A  e.  _I  <->  ( A  e.  ( _V  X.  _V )  /\  ( 1st `  A
)  =  ( 2nd `  A ) ) )

Proof of Theorem bj-elid
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reli 4982 . . . . 5  |-  Rel  _I
2 df-rel 4861 . . . . 5  |-  ( Rel 
_I 
<->  _I  C_  ( _V  X.  _V ) )
31, 2mpbi 211 . . . 4  |-  _I  C_  ( _V  X.  _V )
43sseli 3466 . . 3  |-  ( A  e.  _I  ->  A  e.  ( _V  X.  _V ) )
5 1st2nd2 6844 . . . . . . 7  |-  ( A  e.  ( _V  X.  _V )  ->  A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >. )
64, 5syl 17 . . . . . 6  |-  ( A  e.  _I  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
76eleq1d 2498 . . . . 5  |-  ( A  e.  _I  ->  ( A  e.  _I  <->  <. ( 1st `  A ) ,  ( 2nd `  A )
>.  e.  _I  ) )
87ibi 244 . . . 4  |-  ( A  e.  _I  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  _I  )
9 df-id 4769 . . . . . 6  |-  _I  =  { <. x ,  y
>.  |  x  =  y }
109eleq2i 2507 . . . . 5  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  _I  <->  <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  { <. x ,  y >.  |  x  =  y } )
11 fvex 5891 . . . . . 6  |-  ( 1st `  A )  e.  _V
12 fvex 5891 . . . . . 6  |-  ( 2nd `  A )  e.  _V
13 eqeq12 2448 . . . . . 6  |-  ( ( x  =  ( 1st `  A )  /\  y  =  ( 2nd `  A
) )  ->  (
x  =  y  <->  ( 1st `  A )  =  ( 2nd `  A ) ) )
1411, 12, 13opelopaba 4737 . . . . 5  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  { <. x ,  y >.  |  x  =  y } 
<->  ( 1st `  A
)  =  ( 2nd `  A ) )
1510, 14bitri 252 . . . 4  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  _I  <->  ( 1st `  A )  =  ( 2nd `  A
) )
168, 15sylib 199 . . 3  |-  ( A  e.  _I  ->  ( 1st `  A )  =  ( 2nd `  A
) )
174, 16jca 534 . 2  |-  ( A  e.  _I  ->  ( A  e.  ( _V  X.  _V )  /\  ( 1st `  A )  =  ( 2nd `  A
) ) )
185eleq1d 2498 . . . . 5  |-  ( A  e.  ( _V  X.  _V )  ->  ( A  e.  _I  <->  <. ( 1st `  A ) ,  ( 2nd `  A )
>.  e.  _I  ) )
1918biimprd 226 . . . 4  |-  ( A  e.  ( _V  X.  _V )  ->  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  _I  ->  A  e.  _I  )
)
2015, 19syl5bir 221 . . 3  |-  ( A  e.  ( _V  X.  _V )  ->  ( ( 1st `  A )  =  ( 2nd `  A
)  ->  A  e.  _I  ) )
2120imp 430 . 2  |-  ( ( A  e.  ( _V 
X.  _V )  /\  ( 1st `  A )  =  ( 2nd `  A
) )  ->  A  e.  _I  )
2217, 21impbii 190 1  |-  ( A  e.  _I  <->  ( A  e.  ( _V  X.  _V )  /\  ( 1st `  A
)  =  ( 2nd `  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   _Vcvv 3087    C_ wss 3442   <.cop 4008   {copab 4483    _I cid 4764    X. cxp 4852   Rel wrel 4859   ` cfv 5601   1stc1st 6805   2ndc2nd 6806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-iota 5565  df-fun 5603  df-fv 5609  df-1st 6807  df-2nd 6808
This theorem is referenced by:  bj-elid2  31385  bj-elid3  31386
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