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Theorem bj-elid7 35350
Description: Characterization of the elements of the diagonal of a Cartesian square. (Contributed by BJ, 22-Jun-2019.)
Assertion
Ref Expression
bj-elid7 (⟨𝐵, 𝐶⟩ ∈ ( I ↾ 𝐴) ↔ (𝐵𝐴𝐵 = 𝐶))

Proof of Theorem bj-elid7
StepHypRef Expression
1 df-br 5074 . 2 (𝐵( I ↾ 𝐴)𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ ( I ↾ 𝐴))
2 bj-idreseqb 35342 . 2 (𝐵( I ↾ 𝐴)𝐶 ↔ (𝐵𝐴𝐵 = 𝐶))
31, 2bitr3i 276 1 (⟨𝐵, 𝐶⟩ ∈ ( I ↾ 𝐴) ↔ (𝐵𝐴𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1539  wcel 2106  cop 4567   class class class wbr 5073   I cid 5483  cres 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5221  ax-nul 5228  ax-pr 5350
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3431  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5074  df-opab 5136  df-id 5484  df-xp 5590  df-rel 5591  df-res 5596
This theorem is referenced by: (None)
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