Theorem bj-elid7 37546

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) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶))

Proof of Theorem bj-elid7

Step	Hyp	Ref	Expression
1		df-br 5076	. 2 ⊢ (𝐵( I ↾ 𝐴)𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ ( I ↾ 𝐴))
2		bj-idreseqb 37538	. 2 ⊢ (𝐵( I ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶))
3	1, 2	bitr3i 279	1 ⊢ (⟨𝐵, 𝐶⟩ ∈ ( I ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶))

Syntax hints:   ↔ wb 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ⟨cop 4564   class class class wbr 5075   I cid 5515   ↾ cres 5623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-id 5516  df-xp 5627  df-rel 5628  df-res 5633

This theorem is referenced by: (None)