Theorem resdifcom 5997

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 4235	. 2 ⊢ ((𝐴 ∖ 𝐶) ∩ (𝐵 × V)) = ((𝐴 ∩ (𝐵 × V)) ∖ 𝐶)
2		df-res 5673	. 2 ⊢ ((𝐴 ∖ 𝐶) ↾ 𝐵) = ((𝐴 ∖ 𝐶) ∩ (𝐵 × V))
3		df-res 5673	. . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
4	3	difeq1i 4077	. 2 ⊢ ((𝐴 ↾ 𝐵) ∖ 𝐶) = ((𝐴 ∩ (𝐵 × V)) ∖ 𝐶)
5	1, 2, 4	3eqtr4ri 2797	1 ⊢ ((𝐴 ↾ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ↾ 𝐵)

Syntax hints:   = wceq 1570  Vcvv 3455   ∖ cdif 3902   ∩ cin 3904   × cxp 5659   ↾ cres 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-in 3912  df-res 5673

This theorem is referenced by:  setsfun0  17227  cycpmrn  33463  tocyccntz  33464