Theorem resabs2i 45185

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

Hypothesis

Ref	Expression
resabs2i.1	⊢ 𝐵 ⊆ 𝐶

Assertion

Ref	Expression
resabs2i	⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ 𝐵)

Proof of Theorem resabs2i

Step	Hyp	Ref	Expression
1		resabs2i.1	. 2 ⊢ 𝐵 ⊆ 𝐶
2		resabs2 5957	. 2 ⊢ (𝐵 ⊆ 𝐶 → ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐶) = (𝐴 ↾ 𝐵)

Syntax hints:   = wceq 1541   ⊆ wss 3897   ↾ cres 5616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-opab 5152  df-xp 5620  df-rel 5621  df-res 5626

This theorem is referenced by: (None)