Theorem resabs1i 41782

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 5848	. 2 ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵)

Syntax hints:   = wceq 1538   ⊆ wss 3881   ↾ cres 5521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-opab 5093  df-xp 5525  df-rel 5526  df-res 5531

This theorem is referenced by:  liminfresre  42421