Theorem bj-elid7 37843

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 5109	. 2 ⊢ (𝐵( I ↾ 𝐴)𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ ( I ↾ 𝐴))
2		bj-idreseqb 37835	. 2 ⊢ (𝐵( I ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶))
3	1, 2	bitr3i 280	1 ⊢ (⟨𝐵, 𝐶⟩ ∈ ( I ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 = 𝐶))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ⟨cop 4594   class class class wbr 5108   I cid 5554   ↾ cres 5662

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-id 5555  df-xp 5666  df-rel 5667  df-res 5672

This theorem is used by: (None)