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Theorem bj-eldiag 31159
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 31158 . . 3  |-  ( A  e.  V  ->  (Diag `  A )  =  (  _I  i^i  ( A  X.  A ) ) )
21eleq2d 2472 . 2  |-  ( A  e.  V  ->  ( B  e.  (Diag `  A
)  <->  B  e.  (  _I  i^i  ( A  X.  A ) ) ) )
3 elin 3625 . . 3  |-  ( B  e.  (  _I  i^i  ( A  X.  A
) )  <->  ( B  e.  _I  /\  B  e.  ( A  X.  A
) ) )
4 ancom 448 . . 3  |-  ( ( B  e.  _I  /\  B  e.  ( A  X.  A ) )  <->  ( B  e.  ( A  X.  A
)  /\  B  e.  _I  ) )
5 bj-elid2 31153 . . . 4  |-  ( B  e.  ( A  X.  A )  ->  ( B  e.  _I  <->  ( 1st `  B )  =  ( 2nd `  B ) ) )
65pm5.32i 635 . . 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 367    = wceq 1405    e. wcel 1842    i^i cin 3412    _I cid 4732    X. cxp 4940   ` cfv 5525   1stc1st 6736   2ndc2nd 6737  Diagcdiag2 31156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-iota 5489  df-fun 5527  df-fv 5533  df-1st 6738  df-2nd 6739  df-bj-diag 31157
This theorem is referenced by: (None)
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