Theorem resdifdir 6188

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 4223	. 2 ⊢ ((𝐴 ∖ 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐶 × V)) ∖ (𝐵 ∩ (𝐶 × V)))
2		df-res 5630	. 2 ⊢ ((𝐴 ∖ 𝐵) ↾ 𝐶) = ((𝐴 ∖ 𝐵) ∩ (𝐶 × V))
3		df-res 5630	. . 3 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V))
4		df-res 5630	. . 3 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
5	3, 4	difeq12i 4055	. 2 ⊢ ((𝐴 ↾ 𝐶) ∖ (𝐵 ↾ 𝐶)) = ((𝐴 ∩ (𝐶 × V)) ∖ (𝐵 ∩ (𝐶 × V)))
6	1, 2, 5	3eqtr4i 2772	1 ⊢ ((𝐴 ∖ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∖ (𝐵 ↾ 𝐶))

Syntax hints:   = wceq 1547  Vcvv 3431   ∖ cdif 3880   ∩ cin 3882   × cxp 5616   ↾ cres 5620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-in 3890  df-res 5630

This theorem is referenced by: (None)