Theorem resdifcom 5335

Description: Commutative law for restriction and difference. (Contributed by AV, 7-Jun-2021.)

Assertion

Ref	Expression
resdifcom	⊢ ((𝐴 ↾ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ↾ 𝐵)

Proof of Theorem resdifcom

Step	Hyp	Ref	Expression
1		indif1 3830	. 2 ⊢ ((𝐴 ∖ 𝐶) ∩ (𝐵 × V)) = ((𝐴 ∩ (𝐵 × V)) ∖ 𝐶)
2		df-res 5050	. 2 ⊢ ((𝐴 ∖ 𝐶) ↾ 𝐵) = ((𝐴 ∖ 𝐶) ∩ (𝐵 × V))
3		df-res 5050	. . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
4	3	difeq1i 3686	. 2 ⊢ ((𝐴 ↾ 𝐵) ∖ 𝐶) = ((𝐴 ∩ (𝐵 × V)) ∖ 𝐶)
5	1, 2, 4	3eqtr4ri 2643	1 ⊢ ((𝐴 ↾ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ↾ 𝐵)

Syntax hints:   = wceq 1475  Vcvv 3173   ∖ cdif 3537   ∩ cin 3539   × cxp 5036   ↾ cres 5040

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-in 3547  df-res 5050

This theorem is referenced by:  setsfun0  15726