Theorem resdifdir 6184

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 4242	. 2 ⊢ ((𝐴 ∖ 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐶 × V)) ∖ (𝐵 ∩ (𝐶 × V)))
2		df-res 5626	. 2 ⊢ ((𝐴 ∖ 𝐵) ↾ 𝐶) = ((𝐴 ∖ 𝐵) ∩ (𝐶 × V))
3		df-res 5626	. . 3 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V))
4		df-res 5626	. . 3 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
5	3, 4	difeq12i 4071	. 2 ⊢ ((𝐴 ↾ 𝐶) ∖ (𝐵 ↾ 𝐶)) = ((𝐴 ∩ (𝐶 × V)) ∖ (𝐵 ∩ (𝐶 × V)))
6	1, 2, 5	3eqtr4i 2764	1 ⊢ ((𝐴 ∖ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∖ (𝐵 ↾ 𝐶))

Syntax hints:   = wceq 1541  Vcvv 3436   ∖ cdif 3894   ∩ cin 3896   × cxp 5612   ↾ cres 5616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-in 3904  df-res 5626

This theorem is referenced by: (None)