Theorem bj-dfrel4v 38061

Description: A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 6946 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.) Do not depend on df-cnv 5658, reverse equality. (Revised by BJ, 16-Oct-2026.) Note: this theorem could be used in place of opabid2 5806, 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 38059	. 2 ⊢ (Rel 𝑅 ↔ (𝑅 ↾ V) = 𝑅)
2		relpart 38058	. . 3 ⊢ (𝑅 ↾ V) = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑅𝑦}
3	2	eqeq1i 2765	. 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 1569  Vcvv 3450   class class class wbr 5102  {copab 5166   ↾ cres 5652  Rel wrel 5655

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-in 4090  df-un 4150  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-res 5662

This theorem is used by:  bj-dfrel2  38062