Theorem resdifdi 6139

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 5601	. . 3 ⊢ (𝐴 ↾ (𝐵 ∖ 𝐶)) = (𝐴 ∩ ((𝐵 ∖ 𝐶) × V))
2		difxp1 6068	. . . 4 ⊢ ((𝐵 ∖ 𝐶) × V) = ((𝐵 × V) ∖ (𝐶 × V))
3	2	ineq2i 4143	. . 3 ⊢ (𝐴 ∩ ((𝐵 ∖ 𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∖ (𝐶 × V)))
4		indifdi 4217	. . 3 ⊢ (𝐴 ∩ ((𝐵 × V) ∖ (𝐶 × V))) = ((𝐴 ∩ (𝐵 × V)) ∖ (𝐴 ∩ (𝐶 × V)))
5	1, 3, 4	3eqtri 2770	. 2 ⊢ (𝐴 ↾ (𝐵 ∖ 𝐶)) = ((𝐴 ∩ (𝐵 × V)) ∖ (𝐴 ∩ (𝐶 × V)))
6		df-res 5601	. . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
7		df-res 5601	. . 3 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V))
8	6, 7	difeq12i 4055	. 2 ⊢ ((𝐴 ↾ 𝐵) ∖ (𝐴 ↾ 𝐶)) = ((𝐴 ∩ (𝐵 × V)) ∖ (𝐴 ∩ (𝐶 × V)))
9	5, 8	eqtr4i 2769	1 ⊢ (𝐴 ↾ (𝐵 ∖ 𝐶)) = ((𝐴 ↾ 𝐵) ∖ (𝐴 ↾ 𝐶))

Syntax hints:   = wceq 1539  Vcvv 3432   ∖ cdif 3884   ∩ cin 3886   × cxp 5587   ↾ cres 5591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-res 5601

This theorem is referenced by: (None)