Theorem elrid 6006

Description: Characterization of the elements of a restricted identity relation. (Contributed by BJ, 28-Aug-2022.) (Proof shortened by Peter Mazsa, 9-Sep-2022.)

Assertion

Ref	Expression
elrid	⊢ (𝐴 ∈ ( I ↾ 𝑋) ↔ ∃𝑥 ∈ 𝑋 𝐴 = ⟨𝑥, 𝑥⟩)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑋

Proof of Theorem elrid

Step	Hyp	Ref	Expression
1		df-res 5637	. . 3 ⊢ ( I ↾ 𝑋) = ( I ∩ (𝑋 × V))
2	1	eleq2i 2829	. 2 ⊢ (𝐴 ∈ ( I ↾ 𝑋) ↔ 𝐴 ∈ ( I ∩ (𝑋 × V)))
3		elidinxp 6004	. 2 ⊢ (𝐴 ∈ ( I ∩ (𝑋 × V)) ↔ ∃𝑥 ∈ (𝑋 ∩ V)𝐴 = ⟨𝑥, 𝑥⟩)
4		inv1 4339	. . 3 ⊢ (𝑋 ∩ V) = 𝑋
5	4	rexeqi 3295	. 2 ⊢ (∃𝑥 ∈ (𝑋 ∩ V)𝐴 = ⟨𝑥, 𝑥⟩ ↔ ∃𝑥 ∈ 𝑋 𝐴 = ⟨𝑥, 𝑥⟩)
6	2, 3, 5	3bitri 297	1 ⊢ (𝐴 ∈ ( I ↾ 𝑋) ↔ ∃𝑥 ∈ 𝑋 𝐴 = ⟨𝑥, 𝑥⟩)

Syntax hints:   ↔ wb 206   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  Vcvv 3430   ∩ cin 3889  ⟨cop 4574   I cid 5519   × cxp 5623   ↾ cres 5627

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-id 5520  df-xp 5631  df-rel 5632  df-res 5637

This theorem is referenced by:  idinxpres  6007  idrefALT  6071  elid  6158