Theorem resundi 5944

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 5689	. . . 4 ⊢ ((𝐵 ∪ 𝐶) × V) = ((𝐵 × V) ∪ (𝐶 × V))
2	1	ineq2i 4168	. . 3 ⊢ (𝐴 ∩ ((𝐵 ∪ 𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∪ (𝐶 × V)))
3		indi 4235	. . 3 ⊢ (𝐴 ∩ ((𝐵 × V) ∪ (𝐶 × V))) = ((𝐴 ∩ (𝐵 × V)) ∪ (𝐴 ∩ (𝐶 × V)))
4	2, 3	eqtri 2752	. 2 ⊢ (𝐴 ∩ ((𝐵 ∪ 𝐶) × V)) = ((𝐴 ∩ (𝐵 × V)) ∪ (𝐴 ∩ (𝐶 × V)))
5		df-res 5631	. 2 ⊢ (𝐴 ↾ (𝐵 ∪ 𝐶)) = (𝐴 ∩ ((𝐵 ∪ 𝐶) × V))
6		df-res 5631	. . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
7		df-res 5631	. . 3 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V))
8	6, 7	uneq12i 4117	. 2 ⊢ ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶)) = ((𝐴 ∩ (𝐵 × V)) ∪ (𝐴 ∩ (𝐶 × V)))
9	4, 5, 8	3eqtr4i 2762	1 ⊢ (𝐴 ↾ (𝐵 ∪ 𝐶)) = ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶))

Syntax hints:   = wceq 1540  Vcvv 3436   ∪ cun 3901   ∩ cin 3902   × cxp 5617   ↾ cres 5621

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-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3395  df-v 3438  df-un 3908  df-in 3910  df-opab 5155  df-xp 5625  df-res 5631

This theorem is referenced by:  reldisjun  5983  imaundi  6098  imadifssran  6100  relresfld  6224  resasplit  6694  fresaunres2  6696  residpr  7077  fnsnsplit  7120  eqfunresadj  7297  tfrlem16  8315  mapunen  9063  fnfi  9092  fseq1p1m1  13501  resunimafz0  14352  gsum2dlem2  19850  dprd2da  19923  evlseu  21988  ptuncnv  23692  mbfres2  25544  nosupbnd2lem1  27625  noinfbnd2lem1  27640  ffsrn  32673  resf1o  32674  symgcom  33026  tocyc01  33061  cvmliftlem10  35277  poimirlem9  37619  disjresundif  38227  dvun  42342  eldioph4b  42794  pwssplit4  43072  tfsconcatrev  43331  undmrnresiss  43587  relexp0a  43699  rnresun  45168  tposresg  48872