Theorem resdifcom 5990

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 4263	. 2 ⊢ ((𝐴 ∖ 𝐶) ∩ (𝐵 × V)) = ((𝐴 ∩ (𝐵 × V)) ∖ 𝐶)
2		df-res 5678	. 2 ⊢ ((𝐴 ∖ 𝐶) ↾ 𝐵) = ((𝐴 ∖ 𝐶) ∩ (𝐵 × V))
3		df-res 5678	. . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
4	3	difeq1i 4110	. 2 ⊢ ((𝐴 ↾ 𝐵) ∖ 𝐶) = ((𝐴 ∩ (𝐵 × V)) ∖ 𝐶)
5	1, 2, 4	3eqtr4ri 2763	1 ⊢ ((𝐴 ↾ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ↾ 𝐵)

Syntax hints:   = wceq 1533  Vcvv 3466   ∖ cdif 3937   ∩ cin 3939   × cxp 5664   ↾ cres 5668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3943  df-in 3947  df-res 5678

This theorem is referenced by:  setsfun0  17103  cycpmrn  32736  tocyccntz  32737