Theorem relpart 38186

Description: Two ways of expressing the "relation part" of a class: its restriction to the universal class (that is, its intersection with the Cartesian square of the universal class) equals the class of ordered pairs it contains (the class of ordered pairs whose components are related by it).

The "relation part" of a class is a relation (see df-rel 5665 and relres 6012) and is the largest relation included in that class (that follows from relssresv 38189). Taking the relation part of a class is a projection onto the collection of relations, that is, it is an idempotent operation (residm 6018) with image the collection of relations (by relres 6012 for one inclusion and the forward implication of dfrel3 6211 for the other). As with any projection, a class is in its image if and only if it is a fixpoint of that projection (which is precisely dfrel3 6211).

Other expressions giving the relation part of a class are cnvcnv 6202, coi1res 38199. (Contributed by BJ, 16-Oct-2026.)

Assertion

Ref	Expression
relpart	⊢ (𝐴 ↾ V) = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦}

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem relpart

Step	Hyp	Ref	Expression
1		bj-dfres2 38185	. 2 ⊢ (𝐴 ↾ V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑥𝐴𝑦)}
2		vex 3458	. . . . 5 ⊢ 𝑥 ∈ V
3	2	biantrur 540	. . . 4 ⊢ (𝑥𝐴𝑦 ↔ (𝑥 ∈ V ∧ 𝑥𝐴𝑦))
4	3	bicomi 227	. . 3 ⊢ ((𝑥 ∈ V ∧ 𝑥𝐴𝑦) ↔ 𝑥𝐴𝑦)
5	4	opabbii 5178	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑥𝐴𝑦)} = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦}
6	1, 5	eqtri 2786	1 ⊢ (𝐴 ↾ V) = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝐴𝑦}

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454   class class class wbr 5108  {copab 5173   ↾ 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-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-in 4094  df-un 4154  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-xp 5664  df-res 5670

This theorem is used by:  bj-dfrel4v  38190  bj-cnvcnv2  38192  coi1res  38199  coi2res  38200