Theorem resdifdi 6190

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

Assertion

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

Proof of Theorem resdifdi

Step	Hyp	Ref	Expression
1		df-res 5632	. . 3 ⊢ (𝐴 ↾ (𝐵 ∖ 𝐶)) = (𝐴 ∩ ((𝐵 ∖ 𝐶) × V))
2		difxp1 6119	. . . 4 ⊢ ((𝐵 ∖ 𝐶) × V) = ((𝐵 × V) ∖ (𝐶 × V))
3	2	ineq2i 4148	. . 3 ⊢ (𝐴 ∩ ((𝐵 ∖ 𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∖ (𝐶 × V)))
4		indifdi 4224	. . 3 ⊢ (𝐴 ∩ ((𝐵 × V) ∖ (𝐶 × V))) = ((𝐴 ∩ (𝐵 × V)) ∖ (𝐴 ∩ (𝐶 × V)))
5	1, 3, 4	3eqtri 2768	. 2 ⊢ (𝐴 ↾ (𝐵 ∖ 𝐶)) = ((𝐴 ∩ (𝐵 × V)) ∖ (𝐴 ∩ (𝐶 × V)))
6		df-res 5632	. . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
7		df-res 5632	. . 3 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V))
8	6, 7	difeq12i 4057	. 2 ⊢ ((𝐴 ↾ 𝐵) ∖ (𝐴 ↾ 𝐶)) = ((𝐴 ∩ (𝐵 × V)) ∖ (𝐴 ∩ (𝐶 × V)))
9	5, 8	eqtr4i 2767	1 ⊢ (𝐴 ↾ (𝐵 ∖ 𝐶)) = ((𝐴 ↾ 𝐵) ∖ (𝐴 ↾ 𝐶))

Syntax hints:   = wceq 1548  Vcvv 3433   ∖ cdif 3881   ∩ cin 3883   × cxp 5618   ↾ cres 5622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5220  ax-pr 5364

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-opab 5137  df-xp 5626  df-rel 5627  df-res 5632

This theorem is referenced by: (None)