Theorem resssres0 38179

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

Assertion

Ref	Expression
resssres0	⊢ ((𝐴 ↾ 𝑋) ⊆ 𝐵 ↔ (𝐴 ↾ 𝑋) ⊆ (𝐵 ↾ 𝑋))

Proof of Theorem resssres0

Step	Hyp	Ref	Expression
1		ssid 3935	. 2 ⊢ 𝑋 ⊆ 𝑋
2		resssres 38178	. 2 ⊢ (𝑋 ⊆ 𝑋 → ((𝐴 ↾ 𝑋) ⊆ 𝐵 ↔ (𝐴 ↾ 𝑋) ⊆ (𝐵 ↾ 𝑋)))
3	1, 2	ax-mp 5	1 ⊢ ((𝐴 ↾ 𝑋) ⊆ 𝐵 ↔ (𝐴 ↾ 𝑋) ⊆ (𝐵 ↾ 𝑋))

This proof depends on syntax axioms:   ↔ wb 209   ⊆ wss 3900   ↾ cres 5660

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 5255  ax-pr 5398

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 4841  df-opab 5174  df-xp 5664  df-rel 5665  df-res 5670

This theorem is used by: (None)