Theorem resdifdi 6070

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 5540	. . 3 ⊢ (𝐴 ↾ (𝐵 ∖ 𝐶)) = (𝐴 ∩ ((𝐵 ∖ 𝐶) × V))
2		difxp1 5999	. . . 4 ⊢ ((𝐵 ∖ 𝐶) × V) = ((𝐵 × V) ∖ (𝐶 × V))
3	2	ineq2i 4116	. . 3 ⊢ (𝐴 ∩ ((𝐵 ∖ 𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∖ (𝐶 × V)))
4		indifdi 4190	. . 3 ⊢ (𝐴 ∩ ((𝐵 × V) ∖ (𝐶 × V))) = ((𝐴 ∩ (𝐵 × V)) ∖ (𝐴 ∩ (𝐶 × V)))
5	1, 3, 4	3eqtri 2785	. 2 ⊢ (𝐴 ↾ (𝐵 ∖ 𝐶)) = ((𝐴 ∩ (𝐵 × V)) ∖ (𝐴 ∩ (𝐶 × V)))
6		df-res 5540	. . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
7		df-res 5540	. . 3 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V))
8	6, 7	difeq12i 4028	. 2 ⊢ ((𝐴 ↾ 𝐵) ∖ (𝐴 ↾ 𝐶)) = ((𝐴 ∩ (𝐵 × V)) ∖ (𝐴 ∩ (𝐶 × V)))
9	5, 8	eqtr4i 2784	1 ⊢ (𝐴 ↾ (𝐵 ∖ 𝐶)) = ((𝐴 ↾ 𝐵) ∖ (𝐴 ↾ 𝐶))

Syntax hints:   = wceq 1538  Vcvv 3409   ∖ cdif 3857   ∩ cin 3859   × cxp 5526   ↾ cres 5530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5037  df-opab 5099  df-xp 5534  df-rel 5535  df-cnv 5536  df-res 5540

This theorem is referenced by: (None)