Theorem resdifcom 5665

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 4098	. 2 ⊢ ((𝐴 ∖ 𝐶) ∩ (𝐵 × V)) = ((𝐴 ∩ (𝐵 × V)) ∖ 𝐶)
2		df-res 5367	. 2 ⊢ ((𝐴 ∖ 𝐶) ↾ 𝐵) = ((𝐴 ∖ 𝐶) ∩ (𝐵 × V))
3		df-res 5367	. . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
4	3	difeq1i 3947	. 2 ⊢ ((𝐴 ↾ 𝐵) ∖ 𝐶) = ((𝐴 ∩ (𝐵 × V)) ∖ 𝐶)
5	1, 2, 4	3eqtr4ri 2813	1 ⊢ ((𝐴 ↾ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ↾ 𝐵)

Syntax hints:   = wceq 1601  Vcvv 3398   ∖ cdif 3789   ∩ cin 3791   × cxp 5353   ↾ cres 5357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rab 3099  df-v 3400  df-dif 3795  df-in 3799  df-res 5367

This theorem is referenced by:  setsfun0  16291