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Theorem bj-eldiag 32889
Description: Characterization of the elements the diagonal of a Cartesian square. (Contributed by BJ, 22-Jun-2019.)
Assertion
Ref Expression
bj-eldiag  |-  ( A  e.  V  ->  ( B  e.  (Diag `  A
)  <->  ( B  e.  ( A  X.  A
)  /\  ( 1st `  B )  =  ( 2nd `  B ) ) ) )

Proof of Theorem bj-eldiag
StepHypRef Expression
1 bj-diagval 32888 . . 3  |-  ( A  e.  V  ->  (Diag `  A )  =  (  _I  i^i  ( A  X.  A ) ) )
21eleq2d 2524 . 2  |-  ( A  e.  V  ->  ( B  e.  (Diag `  A
)  <->  B  e.  (  _I  i^i  ( A  X.  A ) ) ) )
3 elin 3650 . . 3  |-  ( B  e.  (  _I  i^i  ( A  X.  A
) )  <->  ( B  e.  _I  /\  B  e.  ( A  X.  A
) ) )
4 ancom 450 . . 3  |-  ( ( B  e.  _I  /\  B  e.  ( A  X.  A ) )  <->  ( B  e.  ( A  X.  A
)  /\  B  e.  _I  ) )
5 bj-elid2 32884 . . . 4  |-  ( B  e.  ( A  X.  A )  ->  ( B  e.  _I  <->  ( 1st `  B )  =  ( 2nd `  B ) ) )
65pm5.32i 637 . . 3  |-  ( ( B  e.  ( A  X.  A )  /\  B  e.  _I  )  <->  ( B  e.  ( A  X.  A )  /\  ( 1st `  B )  =  ( 2nd `  B
) ) )
73, 4, 63bitri 271 . 2  |-  ( B  e.  (  _I  i^i  ( A  X.  A
) )  <->  ( B  e.  ( A  X.  A
)  /\  ( 1st `  B )  =  ( 2nd `  B ) ) )
82, 7syl6bb 261 1  |-  ( A  e.  V  ->  ( B  e.  (Diag `  A
)  <->  ( B  e.  ( A  X.  A
)  /\  ( 1st `  B )  =  ( 2nd `  B ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    i^i cin 3438    _I cid 4742    X. cxp 4949   ` cfv 5529   1stc1st 6688   2ndc2nd 6689  Diagcdiag2 32886
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-pw 3973  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  df-bj-diag 32887
This theorem is referenced by: (None)
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