Theorem elidinxpid 5952

Description: Characterization of the elements of the intersection of the identity relation with a Cartesian square. (Contributed by Peter Mazsa, 9-Sep-2022.)

Assertion

Ref	Expression
elidinxpid	⊢ (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ ∃𝑥 ∈ 𝐴 𝐵 = ⟨𝑥, 𝑥⟩)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem elidinxpid

Step	Hyp	Ref	Expression
1		elidinxp 5951	. 2 ⊢ (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐴)𝐵 = ⟨𝑥, 𝑥⟩)
2		inidm 4152	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
3	2	rexeqi 3347	. 2 ⊢ (∃𝑥 ∈ (𝐴 ∩ 𝐴)𝐵 = ⟨𝑥, 𝑥⟩ ↔ ∃𝑥 ∈ 𝐴 𝐵 = ⟨𝑥, 𝑥⟩)
4	1, 3	bitri 274	1 ⊢ (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ ∃𝑥 ∈ 𝐴 𝐵 = ⟨𝑥, 𝑥⟩)

Syntax hints:   ↔ wb 205   = wceq 1539   ∈ wcel 2106  ∃wrex 3065   ∩ cin 3886  ⟨cop 4567   I cid 5488   × cxp 5587

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 5223  ax-nul 5230  ax-pr 5352

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 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596

This theorem is referenced by: (None)