Theorem resabs1i 39835

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 5585	. 2 ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵)

Syntax hints:   = wceq 1632   ⊆ wss 3715   ↾ cres 5268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-opab 4865  df-xp 5272  df-rel 5273  df-res 5278

This theorem is referenced by:  liminfresre  40514