Theorem elidinxpid 6000

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 5999	. 2 ⊢ (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ ∃𝑥 ∈ (𝐴 ∩ 𝐴)𝐵 = ⟨𝑥, 𝑥⟩)
2		inidm 4180	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
3	2	rexeqi 3289	. 2 ⊢ (∃𝑥 ∈ (𝐴 ∩ 𝐴)𝐵 = ⟨𝑥, 𝑥⟩ ↔ ∃𝑥 ∈ 𝐴 𝐵 = ⟨𝑥, 𝑥⟩)
4	1, 3	bitri 275	1 ⊢ (𝐵 ∈ ( I ∩ (𝐴 × 𝐴)) ↔ ∃𝑥 ∈ 𝐴 𝐵 = ⟨𝑥, 𝑥⟩)

Syntax hints:   ↔ wb 206   = wceq 1540   ∈ wcel 2109  ∃wrex 3053   ∩ cin 3904  ⟨cop 4585   I cid 5517   × cxp 5621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630

This theorem is referenced by: (None)