Theorem resabs2 6013

Description: Absorption law for restriction. (Contributed by NM, 27-Mar-1998.)

Assertion

Ref	Expression
resabs2	⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ 𝐵))

Proof of Theorem resabs2

Step	Hyp	Ref	Expression
1		rescom 6006	. 2 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ↾ 𝐵)
2		resabs1 6010	. 2 ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐶) ↾ 𝐵) = (𝐴 ↾ 𝐵))
3	1, 2	eqtrid 2813	1 ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ 𝐵))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ⊆ wss 3908   ↾ cres 5668

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

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 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-opab 5179  df-xp 5672  df-rel 5673  df-res 5678

This theorem is used by:  residm  6014  fresaunres2  6757  resabs2i  45899  resabs2d  46159  fourierdlem104  46965  fouriersw  46986