Theorem bj-dfrel3 38059

Description: Alternate definition of relation: a class is a relation if and only if it is equal to its "relation part" (see relpart 38058). (Contributed by NM, 14-May-2008.) Do not depend on df-cnv 5658 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		dfss2 4093	. 2 ⊢ (𝑅 ⊆ (V × V) ↔ (𝑅 ∩ (V × V)) = 𝑅)
2		df-rel 5657	. 2 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
3		df-res 5662	. . 3 ⊢ (𝑅 ↾ V) = (𝑅 ∩ (V × V))
4	3	eqeq1i 2765	. 2 ⊢ ((𝑅 ↾ V) = 𝑅 ↔ (𝑅 ∩ (V × V)) = 𝑅)
5	1, 2, 4	3bitr4i 306	1 ⊢ (Rel 𝑅 ↔ (𝑅 ↾ V) = 𝑅)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569  Vcvv 3450   ⊆ wss 3896   ∩ cin 4088   × cxp 5648   ↾ 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

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ss 3897  df-in 4090  df-rel 5657  df-res 5662

This theorem is used by:  relssresv  38060  bj-dfrel4v  38061  bj-coi1  38067