Theorem resundi 5958

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 5701	. . . 4 ⊢ ((𝐵 ∪ 𝐶) × V) = ((𝐵 × V) ∪ (𝐶 × V))
2	1	ineq2i 4157	. . 3 ⊢ (𝐴 ∩ ((𝐵 ∪ 𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∪ (𝐶 × V)))
3		indi 4224	. . 3 ⊢ (𝐴 ∩ ((𝐵 × V) ∪ (𝐶 × V))) = ((𝐴 ∩ (𝐵 × V)) ∪ (𝐴 ∩ (𝐶 × V)))
4	2, 3	eqtri 2759	. 2 ⊢ (𝐴 ∩ ((𝐵 ∪ 𝐶) × V)) = ((𝐴 ∩ (𝐵 × V)) ∪ (𝐴 ∩ (𝐶 × V)))
5		df-res 5643	. 2 ⊢ (𝐴 ↾ (𝐵 ∪ 𝐶)) = (𝐴 ∩ ((𝐵 ∪ 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	uneq12i 4106	. 2 ⊢ ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶)) = ((𝐴 ∩ (𝐵 × V)) ∪ (𝐴 ∩ (𝐶 × V)))
9	4, 5, 8	3eqtr4i 2769	1 ⊢ (𝐴 ↾ (𝐵 ∪ 𝐶)) = ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶))

Syntax hints:   = wceq 1542  Vcvv 3429   ∪ cun 3887   ∩ cin 3888   × cxp 5629   ↾ cres 5633

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-un 3894  df-in 3896  df-opab 5148  df-xp 5637  df-res 5643

This theorem is referenced by:  reldisjun  5997  imaundi  6113  imadifssran  6115  relresfld  6240  resasplit  6710  fresaunres2  6712  residpr  7096  fnsnsplit  7139  eqfunresadj  7315  tfrlem16  8332  mapunen  9084  fnfi  9112  fseq1p1m1  13552  resunimafz0  14407  gsum2dlem2  19946  dprd2da  20019  evlseu  22061  ptuncnv  23772  mbfres2  25612  nosupbnd2lem1  27679  noinfbnd2lem1  27694  ffsrn  32801  resf1o  32803  symgcom  33144  tocyc01  33179  cvmliftlem10  35476  poimirlem9  37950  disjresundif  38567  dvun  42791  eldioph4b  43239  pwssplit4  43517  tfsconcatrev  43776  undmrnresiss  44031  relexp0a  44143  rnresun  45610  tposresg  49353