Theorem resdifdir 6224

Description: Distributive law for restriction over difference. (Contributed by BTernaryTau, 15-Aug-2024.)

Assertion

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

Proof of Theorem resdifdir

Step	Hyp	Ref	Expression
1		indifdir 4247	. 2 ⊢ ((𝐴 ∖ 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐶 × V)) ∖ (𝐵 ∩ (𝐶 × V)))
2		df-res 5659	. 2 ⊢ ((𝐴 ∖ 𝐵) ↾ 𝐶) = ((𝐴 ∖ 𝐵) ∩ (𝐶 × V))
3		df-res 5659	. . 3 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V))
4		df-res 5659	. . 3 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
5	3, 4	difeq12i 4078	. 2 ⊢ ((𝐴 ↾ 𝐶) ∖ (𝐵 ↾ 𝐶)) = ((𝐴 ∩ (𝐶 × V)) ∖ (𝐵 ∩ (𝐶 × V)))
6	1, 2, 5	3eqtr4i 2795	1 ⊢ ((𝐴 ∖ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∖ (𝐵 ↾ 𝐶))

Syntax hints:   = wceq 1560  Vcvv 3454   ∖ cdif 3901   ∩ cin 3903   × cxp 5645   ↾ cres 5649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-dif 3907  df-in 3911  df-res 5659

This theorem is referenced by: (None)