Theorem elrid 6056

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 5670	. . 3 ⊢ ( I ↾ 𝑋) = ( I ∩ (𝑋 × V))
2	1	eleq2i 2856	. 2 ⊢ (𝐴 ∈ ( I ↾ 𝑋) ↔ 𝐴 ∈ ( I ∩ (𝑋 × V)))
3		elidinxp 6054	. 2 ⊢ (𝐴 ∈ ( I ∩ (𝑋 × V)) ↔ ∃𝑥 ∈ (𝑋 ∩ V)𝐴 = ⟨𝑥, 𝑥⟩)
4		inv1 4351	. . 3 ⊢ (𝑋 ∩ V) = 𝑋
5	4	rexeqi 3322	. 2 ⊢ (∃𝑥 ∈ (𝑋 ∩ V)𝐴 = ⟨𝑥, 𝑥⟩ ↔ ∃𝑥 ∈ 𝑋 𝐴 = ⟨𝑥, 𝑥⟩)
6	2, 3, 5	3bitri 300	1 ⊢ (𝐴 ∈ ( I ↾ 𝑋) ↔ ∃𝑥 ∈ 𝑋 𝐴 = ⟨𝑥, 𝑥⟩)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2147  ∃wrex 3090  Vcvv 3454   ∩ cin 4092  ⟨cop 4840   I cid 5552   × cxp 5656   ↾ cres 5660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-res 5670

This theorem is used by:  idinxpres  6057  idrefALT  6125  elid  6212  idresdmsscocnv1  6281