Theorem resssres 38138

Description: A restricted class is included in a class if and only if it is included in that class restricted to any class including the first restricting class. (Contributed by BJ, 25-Oct-2026.)

Assertion

Ref	Expression
resssres	⊢ (𝑋 ⊆ 𝑌 → ((𝐴 ↾ 𝑋) ⊆ 𝐵 ↔ (𝐴 ↾ 𝑋) ⊆ (𝐵 ↾ 𝑌)))

Proof of Theorem resssres

Step	Hyp	Ref	Expression
1		ssres 6004	. . 3 ⊢ ((𝐴 ↾ 𝑋) ⊆ 𝐵 → ((𝐴 ↾ 𝑋) ↾ 𝑌) ⊆ (𝐵 ↾ 𝑌))
2		resabs2 6011	. . . 4 ⊢ (𝑋 ⊆ 𝑌 → ((𝐴 ↾ 𝑋) ↾ 𝑌) = (𝐴 ↾ 𝑋))
3	2	sseq1d 3944	. . 3 ⊢ (𝑋 ⊆ 𝑌 → (((𝐴 ↾ 𝑋) ↾ 𝑌) ⊆ (𝐵 ↾ 𝑌) ↔ (𝐴 ↾ 𝑋) ⊆ (𝐵 ↾ 𝑌)))
4	1, 3	imbitrid 247	. 2 ⊢ (𝑋 ⊆ 𝑌 → ((𝐴 ↾ 𝑋) ⊆ 𝐵 → (𝐴 ↾ 𝑋) ⊆ (𝐵 ↾ 𝑌)))
5		id 23	. . 3 ⊢ ((𝐴 ↾ 𝑋) ⊆ (𝐵 ↾ 𝑌) → (𝐴 ↾ 𝑋) ⊆ (𝐵 ↾ 𝑌))
6		resss 6002	. . 3 ⊢ (𝐵 ↾ 𝑌) ⊆ 𝐵
7	5, 6	sstrdi 3925	. 2 ⊢ ((𝐴 ↾ 𝑋) ⊆ (𝐵 ↾ 𝑌) → (𝐴 ↾ 𝑋) ⊆ 𝐵)
8	4, 7	impbid1 228	1 ⊢ (𝑋 ⊆ 𝑌 → ((𝐴 ↾ 𝑋) ⊆ 𝐵 ↔ (𝐴 ↾ 𝑋) ⊆ (𝐵 ↾ 𝑌)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ⊆ wss 3900   ↾ cres 5657

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  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-opab 5171  df-xp 5661  df-rel 5662  df-res 5667

This theorem is used by: (None)