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Theorem bj-elid3 34744
Description: Characterization of the elements of  _I. (Contributed by BJ, 29-Mar-2020.)
Assertion
Ref Expression
bj-elid3  |-  ( <. A ,  B >.  e.  _I  <->  ( A  e. 
_V  /\  A  =  B ) )

Proof of Theorem bj-elid3
StepHypRef Expression
1 bj-elid 34742 . 2  |-  ( <. A ,  B >.  e.  _I  <->  ( <. A ,  B >.  e.  ( _V 
X.  _V )  /\  ( 1st `  <. A ,  B >. )  =  ( 2nd `  <. A ,  B >. ) ) )
2 opelxp 5038 . . . 4  |-  ( <. A ,  B >.  e.  ( _V  X.  _V ) 
<->  ( A  e.  _V  /\  B  e.  _V )
)
32anbi1i 695 . . 3  |-  ( (
<. A ,  B >.  e.  ( _V  X.  _V )  /\  ( 1st `  <. A ,  B >. )  =  ( 2nd `  <. A ,  B >. )
)  <->  ( ( A  e.  _V  /\  B  e.  _V )  /\  ( 1st `  <. A ,  B >. )  =  ( 2nd `  <. A ,  B >. ) ) )
4 op1stg 6811 . . . . . 6  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( 1st `  <. A ,  B >. )  =  A )
5 op2ndg 6812 . . . . . 6  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( 2nd `  <. A ,  B >. )  =  B )
64, 5eqeq12d 2479 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( ( 1st `  <. A ,  B >. )  =  ( 2nd `  <. A ,  B >. )  <->  A  =  B ) )
76pm5.32i 637 . . . 4  |-  ( ( ( A  e.  _V  /\  B  e.  _V )  /\  ( 1st `  <. A ,  B >. )  =  ( 2nd `  <. A ,  B >. )
)  <->  ( ( A  e.  _V  /\  B  e.  _V )  /\  A  =  B ) )
8 simpl 457 . . . . . 6  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  A  e.  _V )
98anim1i 568 . . . . 5  |-  ( ( ( A  e.  _V  /\  B  e.  _V )  /\  A  =  B
)  ->  ( A  e.  _V  /\  A  =  B ) )
10 simpl 457 . . . . . 6  |-  ( ( A  e.  _V  /\  A  =  B )  ->  A  e.  _V )
11 eleq1 2529 . . . . . . 7  |-  ( A  =  B  ->  ( A  e.  _V  <->  B  e.  _V ) )
1211biimpac 486 . . . . . 6  |-  ( ( A  e.  _V  /\  A  =  B )  ->  B  e.  _V )
13 simpr 461 . . . . . 6  |-  ( ( A  e.  _V  /\  A  =  B )  ->  A  =  B )
1410, 12, 13jca31 534 . . . . 5  |-  ( ( A  e.  _V  /\  A  =  B )  ->  ( ( A  e. 
_V  /\  B  e.  _V )  /\  A  =  B ) )
159, 14impbii 188 . . . 4  |-  ( ( ( A  e.  _V  /\  B  e.  _V )  /\  A  =  B
)  <->  ( A  e. 
_V  /\  A  =  B ) )
167, 15bitri 249 . . 3  |-  ( ( ( A  e.  _V  /\  B  e.  _V )  /\  ( 1st `  <. A ,  B >. )  =  ( 2nd `  <. A ,  B >. )
)  <->  ( A  e. 
_V  /\  A  =  B ) )
173, 16bitri 249 . 2  |-  ( (
<. A ,  B >.  e.  ( _V  X.  _V )  /\  ( 1st `  <. A ,  B >. )  =  ( 2nd `  <. A ,  B >. )
)  <->  ( A  e. 
_V  /\  A  =  B ) )
181, 17bitri 249 1  |-  ( <. A ,  B >.  e.  _I  <->  ( A  e. 
_V  /\  A  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   _Vcvv 3109   <.cop 4038    _I cid 4799    X. cxp 5006   ` cfv 5594   1stc1st 6797   2ndc2nd 6798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fv 5602  df-1st 6799  df-2nd 6800
This theorem is referenced by:  bj-eldiag2  34750
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