Theorem bj-dfrel3 38187

Description: Alternate definition of relation: a class is a relation if and only if it is equal to its "relation part" (see relpart 38186). (Contributed by NM, 14-May-2008.) Do not depend on df-cnv 5666 to move it earlier. (Revised by BJ, 16-Oct-2026.)

Assertion

Ref	Expression
bj-dfrel3	⊢ (Rel 𝑅 ↔ (𝑅 ↾ V) = 𝑅)

Proof of Theorem bj-dfrel3

Step	Hyp	Ref	Expression
1		df-rel 5665	. 2 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
2		bj-resssxp 38183	. 2 ⊢ ((𝑅 ↾ V) = 𝑅 ↔ 𝑅 ⊆ (V × V))
3	1, 2	bitr4i 281	1 ⊢ (Rel 𝑅 ↔ (𝑅 ↾ V) = 𝑅)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570  Vcvv 3454   ⊆ wss 3900   × cxp 5656   ↾ 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

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ss 3901  df-in 4094  df-rel 5665  df-res 5670

This theorem is used by:  relssresv  38189  bj-dfrel4v  38190  bj-coi1  38203