Theorem relssresv 38189

Description: A relation is included in a class if and only if it is included in its "relation part" (see relpart 38186). (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 38187	. 2 ⊢ (Rel 𝐵 ↔ (𝐵 ↾ V) = 𝐵)
2		ssres 6010	. . . 4 ⊢ (𝐵 ⊆ 𝐴 → (𝐵 ↾ V) ⊆ (𝐴 ↾ V))
3		sseq1 3938	. . . 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 6008	. . . 4 ⊢ (𝐴 ↾ V) ⊆ 𝐴
7	5, 6	sstrdi 3925	. . 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 1570  Vcvv 3454   ⊆ wss 3900   ↾ 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-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-in 4094  df-rel 5665  df-res 5670

This theorem is used by: (None)