Theorem elrid 6050

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 5675	. . 3 ⊢ ( I ↾ 𝑋) = ( I ∩ (𝑋 × V))
2	1	eleq2i 2857	. 2 ⊢ (𝐴 ∈ ( I ↾ 𝑋) ↔ 𝐴 ∈ ( I ∩ (𝑋 × V)))
3		elidinxp 6048	. 2 ⊢ (𝐴 ∈ ( I ∩ (𝑋 × V)) ↔ ∃𝑥 ∈ (𝑋 ∩ V)𝐴 = ⟨𝑥, 𝑥⟩)
4		inv1 4355	. . 3 ⊢ (𝑋 ∩ V) = 𝑋
5	4	rexeqi 3324	. 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 2146  ∃wrex 3091  Vcvv 3457   ∩ cin 3905  ⟨cop 4597   I cid 5557   × cxp 5661   ↾ cres 5665

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

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 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-res 5675

This theorem is used by:  idinxpres  6051  idrefALT  6115  elid  6200