Theorem inres2 39445

Description: Two ways of expressing the restriction of an intersection. (Contributed by Peter Mazsa, 5-Jun-2021.)

Assertion

Ref	Expression
inres2	⊢ ((𝑅 ↾ 𝐴) ∩ 𝑆) = ((𝑅 ∩ 𝑆) ↾ 𝐴)

Proof of Theorem inres2

Step	Hyp	Ref	Expression
1		inres 6004	. . 3 ⊢ (𝑆 ∩ (𝑅 ↾ 𝐴)) = ((𝑆 ∩ 𝑅) ↾ 𝐴)
2	1	ineqcomi 4111	. 2 ⊢ ((𝑅 ↾ 𝐴) ∩ 𝑆) = ((𝑆 ∩ 𝑅) ↾ 𝐴)
3		incom 4109	. . 3 ⊢ (𝑅 ∩ 𝑆) = (𝑆 ∩ 𝑅)
4	3	reseq1i 5982	. 2 ⊢ ((𝑅 ∩ 𝑆) ↾ 𝐴) = ((𝑆 ∩ 𝑅) ↾ 𝐴)
5	2, 4	eqtr4i 2789	1 ⊢ ((𝑅 ↾ 𝐴) ∩ 𝑆) = ((𝑅 ∩ 𝑆) ↾ 𝐴)

This proof depends on syntax axioms:   = wceq 1570   ∩ cin 4092   ↾ cres 5660

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-in 4094  df-res 5670

This theorem is used by:  xrnres  39623