Theorem resabs1i 5989

Description: Absorption law for restriction. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypothesis

Ref	Expression
resabs1i.1	⊢ 𝐵 ⊆ 𝐶

Assertion

Ref	Expression
resabs1i	⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵)

Proof of Theorem resabs1i

Step	Hyp	Ref	Expression
1		resabs1i.1	. 2 ⊢ 𝐵 ⊆ 𝐶
2		resabs1 5988	. 2 ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵)

Syntax hints:   = wceq 1559   ⊆ wss 3902   ↾ cres 5645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-opab 5160  df-xp 5649  df-rel 5650  df-res 5655

This theorem is referenced by:  resindm  6012  resf1extb  7910  liminfresre  46314