Theorem inres2 38203

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 6029	. . 3 ⊢ (𝑆 ∩ (𝑅 ↾ 𝐴)) = ((𝑆 ∩ 𝑅) ↾ 𝐴)
2	1	ineqcomi 4232	. 2 ⊢ ((𝑅 ↾ 𝐴) ∩ 𝑆) = ((𝑆 ∩ 𝑅) ↾ 𝐴)
3		incom 4230	. . 3 ⊢ (𝑅 ∩ 𝑆) = (𝑆 ∩ 𝑅)
4	3	reseq1i 6007	. 2 ⊢ ((𝑅 ∩ 𝑆) ↾ 𝐴) = ((𝑆 ∩ 𝑅) ↾ 𝐴)
5	2, 4	eqtr4i 2771	1 ⊢ ((𝑅 ↾ 𝐴) ∩ 𝑆) = ((𝑅 ∩ 𝑆) ↾ 𝐴)

Syntax hints:   = wceq 1537   ∩ cin 3975   ↾ cres 5702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-in 3983  df-res 5712

This theorem is referenced by:  xrnres  38360