Theorem inres2 38614

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 5949	. . 3 ⊢ (𝑆 ∩ (𝑅 ↾ 𝐴)) = ((𝑆 ∩ 𝑅) ↾ 𝐴)
2	1	ineqcomi 4140	. 2 ⊢ ((𝑅 ↾ 𝐴) ∩ 𝑆) = ((𝑆 ∩ 𝑅) ↾ 𝐴)
3		incom 4138	. . 3 ⊢ (𝑅 ∩ 𝑆) = (𝑆 ∩ 𝑅)
4	3	reseq1i 5927	. 2 ⊢ ((𝑅 ∩ 𝑆) ↾ 𝐴) = ((𝑆 ∩ 𝑅) ↾ 𝐴)
5	2, 4	eqtr4i 2765	1 ⊢ ((𝑅 ↾ 𝐴) ∩ 𝑆) = ((𝑅 ∩ 𝑆) ↾ 𝐴)

Syntax hints:   = wceq 1547   ∩ cin 3882   ↾ cres 5620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-in 3890  df-res 5630

This theorem is referenced by:  xrnres  38792