Theorem relssresv 38060

Description: A relation is included in a class if and only if it is included in its "relation part" (see relpart 38058). (Contributed by BJ, 16-Oct-2026.)

Assertion

Ref	Expression
relssresv	⊢ (Rel 𝐵 → (𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ (𝐴 ↾ V)))

Proof of Theorem relssresv

Step	Hyp	Ref	Expression
1		bj-dfrel3 38059	. 2 ⊢ (Rel 𝐵 ↔ (𝐵 ↾ V) = 𝐵)
2		ssres 5994	. . . 4 ⊢ (𝐵 ⊆ 𝐴 → (𝐵 ↾ V) ⊆ (𝐴 ↾ V))
3		sseq1 3934	. . . 4 ⊢ ((𝐵 ↾ V) = 𝐵 → ((𝐵 ↾ V) ⊆ (𝐴 ↾ V) ↔ 𝐵 ⊆ (𝐴 ↾ V)))
4	2, 3	imbitrid 247	. . 3 ⊢ ((𝐵 ↾ V) = 𝐵 → (𝐵 ⊆ 𝐴 → 𝐵 ⊆ (𝐴 ↾ V)))
5		id 23	. . . 4 ⊢ (𝐵 ⊆ (𝐴 ↾ V) → 𝐵 ⊆ (𝐴 ↾ V))
6		resss 5992	. . . 4 ⊢ (𝐴 ↾ V) ⊆ 𝐴
7	5, 6	sstrdi 3921	. . 3 ⊢ (𝐵 ⊆ (𝐴 ↾ V) → 𝐵 ⊆ 𝐴)
8	4, 7	impbid1 228	. 2 ⊢ ((𝐵 ↾ V) = 𝐵 → (𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ (𝐴 ↾ V)))
9	1, 8	sylbi 220	1 ⊢ (Rel 𝐵 → (𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ (𝐴 ↾ V)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1569  Vcvv 3450   ⊆ wss 3896   ↾ 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-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3897  df-in 4090  df-rel 5657  df-res 5662

This theorem is used by: (None)