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Theorem bj-elid7 34501
 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 5054 . 2 (𝐵( I ↾ 𝐴)𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ ( I ↾ 𝐴))
2 bj-idreseqb 34493 . 2 (𝐵( I ↾ 𝐴)𝐶 ↔ (𝐵𝐴𝐵 = 𝐶))
31, 2bitr3i 280 1 (⟨𝐵, 𝐶⟩ ∈ ( I ↾ 𝐴) ↔ (𝐵𝐴𝐵 = 𝐶))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115  ⟨cop 4556   class class class wbr 5053   I cid 5447   ↾ cres 5545 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5054  df-opab 5116  df-id 5448  df-xp 5549  df-rel 5550  df-res 5555 This theorem is referenced by: (None)
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