Theorem resdifdir 6129

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 4215	. 2 ⊢ ((𝐴 ∖ 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐶 × V)) ∖ (𝐵 ∩ (𝐶 × V)))
2		df-res 5592	. 2 ⊢ ((𝐴 ∖ 𝐵) ↾ 𝐶) = ((𝐴 ∖ 𝐵) ∩ (𝐶 × V))
3		df-res 5592	. . 3 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V))
4		df-res 5592	. . 3 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
5	3, 4	difeq12i 4051	. 2 ⊢ ((𝐴 ↾ 𝐶) ∖ (𝐵 ↾ 𝐶)) = ((𝐴 ∩ (𝐶 × V)) ∖ (𝐵 ∩ (𝐶 × V)))
6	1, 2, 5	3eqtr4i 2776	1 ⊢ ((𝐴 ∖ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∖ (𝐵 ↾ 𝐶))

Syntax hints:   = wceq 1539  Vcvv 3422   ∖ cdif 3880   ∩ cin 3882   × cxp 5578   ↾ cres 5582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-in 3890  df-res 5592

This theorem is referenced by: (None)