Theorem bj-dfrel4v 38190

Description: A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6974 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.) Do not depend on df-cnv 5666, reverse equality. (Revised by BJ, 16-Oct-2026.) Note: this theorem could be used in place of opabid2 5816, which could be deleted.

Assertion

Ref	Expression
bj-dfrel4v	⊢ (Rel 𝑅 ↔ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} = 𝑅)

Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem bj-dfrel4v

Step	Hyp	Ref	Expression
1		bj-dfrel3 38187	. 2 ⊢ (Rel 𝑅 ↔ (𝑅 ↾ V) = 𝑅)
2		relpart 38186	. . 3 ⊢ (𝑅 ↾ V) = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
3	2	eqeq1i 2768	. 2 ⊢ ((𝑅 ↾ V) = 𝑅 ↔ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} = 𝑅)
4	1, 3	bitri 278	1 ⊢ (Rel 𝑅 ↔ {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦} = 𝑅)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570  Vcvv 3454   class class class wbr 5108  {copab 5173   ↾ cres 5660  Rel wrel 5663

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-rel 5665  df-res 5670

This theorem is used by:  bj-dfrel2  38191