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Theorem bj-elid 33672
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 5128 . . . . 5  |-  Rel  _I
2 df-rel 5006 . . . . 5  |-  ( Rel 
_I 
<->  _I  C_  ( _V  X.  _V ) )
31, 2mpbi 208 . . . 4  |-  _I  C_  ( _V  X.  _V )
43sseli 3500 . . 3  |-  ( A  e.  _I  ->  A  e.  ( _V  X.  _V ) )
5 1st2nd2 6818 . . . . . . 7  |-  ( A  e.  ( _V  X.  _V )  ->  A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >. )
64, 5syl 16 . . . . . 6  |-  ( A  e.  _I  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
76eleq1d 2536 . . . . 5  |-  ( A  e.  _I  ->  ( A  e.  _I  <->  <. ( 1st `  A ) ,  ( 2nd `  A )
>.  e.  _I  ) )
87ibi 241 . . . 4  |-  ( A  e.  _I  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  _I  )
9 df-id 4795 . . . . . 6  |-  _I  =  { <. x ,  y
>.  |  x  =  y }
109eleq2i 2545 . . . . 5  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  _I  <->  <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  { <. x ,  y >.  |  x  =  y } )
11 fvex 5874 . . . . . 6  |-  ( 1st `  A )  e.  _V
12 fvex 5874 . . . . . 6  |-  ( 2nd `  A )  e.  _V
13 eqeq12 2486 . . . . . 6  |-  ( ( x  =  ( 1st `  A )  /\  y  =  ( 2nd `  A
) )  ->  (
x  =  y  <->  ( 1st `  A )  =  ( 2nd `  A ) ) )
1411, 12, 13opelopaba 4763 . . . . 5  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  { <. x ,  y >.  |  x  =  y } 
<->  ( 1st `  A
)  =  ( 2nd `  A ) )
1510, 14bitri 249 . . . 4  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  _I  <->  ( 1st `  A )  =  ( 2nd `  A
) )
168, 15sylib 196 . . 3  |-  ( A  e.  _I  ->  ( 1st `  A )  =  ( 2nd `  A
) )
174, 16jca 532 . 2  |-  ( A  e.  _I  ->  ( A  e.  ( _V  X.  _V )  /\  ( 1st `  A )  =  ( 2nd `  A
) ) )
185eleq1d 2536 . . . . 5  |-  ( A  e.  ( _V  X.  _V )  ->  ( A  e.  _I  <->  <. ( 1st `  A ) ,  ( 2nd `  A )
>.  e.  _I  ) )
1918biimprd 223 . . . 4  |-  ( A  e.  ( _V  X.  _V )  ->  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  _I  ->  A  e.  _I  )
)
2015, 19syl5bir 218 . . 3  |-  ( A  e.  ( _V  X.  _V )  ->  ( ( 1st `  A )  =  ( 2nd `  A
)  ->  A  e.  _I  ) )
2120imp 429 . 2  |-  ( ( A  e.  ( _V 
X.  _V )  /\  ( 1st `  A )  =  ( 2nd `  A
) )  ->  A  e.  _I  )
2217, 21impbii 188 1  |-  ( A  e.  _I  <->  ( A  e.  ( _V  X.  _V )  /\  ( 1st `  A
)  =  ( 2nd `  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113    C_ wss 3476   <.cop 4033   {copab 4504    _I cid 4790    X. cxp 4997   Rel wrel 5004   ` cfv 5586   1stc1st 6779   2ndc2nd 6780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5549  df-fun 5588  df-fv 5594  df-1st 6781  df-2nd 6782
This theorem is referenced by:  bj-elid2  33673
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