Theorem resundi 5894

Description: Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)

Assertion

Ref	Expression
resundi	⊢ (𝐴 ↾ (𝐵 ∪ 𝐶)) = ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶))

Proof of Theorem resundi

Step	Hyp	Ref	Expression
1		xpundir 5647	. . . 4 ⊢ ((𝐵 ∪ 𝐶) × V) = ((𝐵 × V) ∪ (𝐶 × V))
2	1	ineq2i 4140	. . 3 ⊢ (𝐴 ∩ ((𝐵 ∪ 𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∪ (𝐶 × V)))
3		indi 4204	. . 3 ⊢ (𝐴 ∩ ((𝐵 × V) ∪ (𝐶 × V))) = ((𝐴 ∩ (𝐵 × V)) ∪ (𝐴 ∩ (𝐶 × V)))
4	2, 3	eqtri 2766	. 2 ⊢ (𝐴 ∩ ((𝐵 ∪ 𝐶) × V)) = ((𝐴 ∩ (𝐵 × V)) ∪ (𝐴 ∩ (𝐶 × V)))
5		df-res 5592	. 2 ⊢ (𝐴 ↾ (𝐵 ∪ 𝐶)) = (𝐴 ∩ ((𝐵 ∪ 𝐶) × V))
6		df-res 5592	. . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
7		df-res 5592	. . 3 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V))
8	6, 7	uneq12i 4091	. 2 ⊢ ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶)) = ((𝐴 ∩ (𝐵 × V)) ∪ (𝐴 ∩ (𝐶 × V)))
9	4, 5, 8	3eqtr4i 2776	1 ⊢ (𝐴 ↾ (𝐵 ∪ 𝐶)) = ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶))

Syntax hints:   = wceq 1539  Vcvv 3422   ∪ cun 3881   ∩ cin 3882   × cxp 5578   ↾ cres 5582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-un 3888  df-in 3890  df-opab 5133  df-xp 5586  df-res 5592

This theorem is referenced by:  imaundi  6042  relresfld  6168  resasplit  6628  fresaunres2  6630  residpr  6997  fnsnsplit  7038  tfrlem16  8195  mapunen  8882  fnfi  8925  fseq1p1m1  13259  resunimafz0  14085  gsum2dlem2  19487  dprd2da  19560  evlseu  21203  ptuncnv  22866  mbfres2  24714  reldisjun  30843  ffsrn  30966  resf1o  30967  symgcom  31254  tocyc01  31287  cvmliftlem10  33156  eqfunresadj  33641  nosupbnd2lem1  33845  noinfbnd2lem1  33860  poimirlem9  35713  eldioph4b  40549  pwssplit4  40830  undmrnresiss  41101  relexp0a  41213  rnresun  42605