Theorem resdifdi 6197

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 5643	. . 3 ⊢ (𝐴 ↾ (𝐵 ∖ 𝐶)) = (𝐴 ∩ ((𝐵 ∖ 𝐶) × V))
2		difxp1 6126	. . . 4 ⊢ ((𝐵 ∖ 𝐶) × V) = ((𝐵 × V) ∖ (𝐶 × V))
3	2	ineq2i 4176	. . 3 ⊢ (𝐴 ∩ ((𝐵 ∖ 𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∖ (𝐶 × V)))
4		indifdi 4253	. . 3 ⊢ (𝐴 ∩ ((𝐵 × V) ∖ (𝐶 × V))) = ((𝐴 ∩ (𝐵 × V)) ∖ (𝐴 ∩ (𝐶 × V)))
5	1, 3, 4	3eqtri 2756	. 2 ⊢ (𝐴 ↾ (𝐵 ∖ 𝐶)) = ((𝐴 ∩ (𝐵 × V)) ∖ (𝐴 ∩ (𝐶 × V)))
6		df-res 5643	. . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
7		df-res 5643	. . 3 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V))
8	6, 7	difeq12i 4083	. 2 ⊢ ((𝐴 ↾ 𝐵) ∖ (𝐴 ↾ 𝐶)) = ((𝐴 ∩ (𝐵 × V)) ∖ (𝐴 ∩ (𝐶 × V)))
9	5, 8	eqtr4i 2755	1 ⊢ (𝐴 ↾ (𝐵 ∖ 𝐶)) = ((𝐴 ↾ 𝐵) ∖ (𝐴 ↾ 𝐶))

Syntax hints:   = wceq 1540  Vcvv 3444   ∖ cdif 3908   ∩ cin 3910   × cxp 5629   ↾ cres 5633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-xp 5637  df-rel 5638  df-cnv 5639  df-res 5643

This theorem is referenced by: (None)