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Theorem bj-elid 32878
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 5078 . . . . 5  |-  Rel  _I
2 df-rel 4958 . . . . 5  |-  ( Rel 
_I 
<->  _I  C_  ( _V  X.  _V ) )
31, 2mpbi 208 . . . 4  |-  _I  C_  ( _V  X.  _V )
43sseli 3463 . . 3  |-  ( A  e.  _I  ->  A  e.  ( _V  X.  _V ) )
5 1st2nd2 6726 . . . . . . 7  |-  ( A  e.  ( _V  X.  _V )  ->  A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >. )
64, 5syl 16 . . . . . 6  |-  ( A  e.  _I  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
76eleq1d 2523 . . . . 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 4747 . . . . . 6  |-  _I  =  { <. x ,  y
>.  |  x  =  y }
109eleq2i 2532 . . . . 5  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  _I  <->  <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  { <. x ,  y >.  |  x  =  y } )
11 fvex 5812 . . . . . 6  |-  ( 1st `  A )  e.  _V
12 fvex 5812 . . . . . 6  |-  ( 2nd `  A )  e.  _V
13 eqeq12 2473 . . . . . 6  |-  ( ( x  =  ( 1st `  A )  /\  y  =  ( 2nd `  A
) )  ->  (
x  =  y  <->  ( 1st `  A )  =  ( 2nd `  A ) ) )
1411, 12, 13opelopaba 4716 . . . . 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 2523 . . . . 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 1370    e. wcel 1758   _Vcvv 3078    C_ wss 3439   <.cop 3994   {copab 4460    _I cid 4742    X. cxp 4949   Rel wrel 4956   ` cfv 5529   1stc1st 6688   2ndc2nd 6689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-iota 5492  df-fun 5531  df-fv 5537  df-1st 6690  df-2nd 6691
This theorem is referenced by:  bj-elid2  32879
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