Theorem resundi 4790

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 4556	. . . 4 ⊢ ((𝐵 ∪ 𝐶) × V) = ((𝐵 × V) ∪ (𝐶 × V))
2	1	ineq2i 3240	. . 3 ⊢ (𝐴 ∩ ((𝐵 ∪ 𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∪ (𝐶 × V)))
3		indi 3289	. . 3 ⊢ (𝐴 ∩ ((𝐵 × V) ∪ (𝐶 × V))) = ((𝐴 ∩ (𝐵 × V)) ∪ (𝐴 ∩ (𝐶 × V)))
4	2, 3	eqtri 2135	. 2 ⊢ (𝐴 ∩ ((𝐵 ∪ 𝐶) × V)) = ((𝐴 ∩ (𝐵 × V)) ∪ (𝐴 ∩ (𝐶 × V)))
5		df-res 4511	. 2 ⊢ (𝐴 ↾ (𝐵 ∪ 𝐶)) = (𝐴 ∩ ((𝐵 ∪ 𝐶) × V))
6		df-res 4511	. . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
7		df-res 4511	. . 3 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V))
8	6, 7	uneq12i 3194	. 2 ⊢ ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶)) = ((𝐴 ∩ (𝐵 × V)) ∪ (𝐴 ∩ (𝐶 × V)))
9	4, 5, 8	3eqtr4i 2145	1 ⊢ (𝐴 ↾ (𝐵 ∪ 𝐶)) = ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶))

Syntax hints:   = wceq 1314  Vcvv 2657   ∪ cun 3035   ∩ cin 3036   × cxp 4497   ↾ cres 4501

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-un 3041  df-in 3043  df-opab 3950  df-xp 4505  df-res 4511

This theorem is referenced by:  imaundi  4909  relresfld  5026  relcoi1  5028  resasplitss  5260  fnsnsplitss  5573  fnsnsplitdc  6355  fnfi  6777  fseq1p1m1  9767  resunimafz0  10467