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Mirrors > Home > ILE Home > Th. List > resundi | GIF version |
Description: Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.) |
Ref | Expression |
---|---|
resundi | ⊢ (𝐴 ↾ (𝐵 ∪ 𝐶)) = ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpundir 4596 | . . . 4 ⊢ ((𝐵 ∪ 𝐶) × V) = ((𝐵 × V) ∪ (𝐶 × V)) | |
2 | 1 | ineq2i 3274 | . . 3 ⊢ (𝐴 ∩ ((𝐵 ∪ 𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∪ (𝐶 × V))) |
3 | indi 3323 | . . 3 ⊢ (𝐴 ∩ ((𝐵 × V) ∪ (𝐶 × V))) = ((𝐴 ∩ (𝐵 × V)) ∪ (𝐴 ∩ (𝐶 × V))) | |
4 | 2, 3 | eqtri 2160 | . 2 ⊢ (𝐴 ∩ ((𝐵 ∪ 𝐶) × V)) = ((𝐴 ∩ (𝐵 × V)) ∪ (𝐴 ∩ (𝐶 × V))) |
5 | df-res 4551 | . 2 ⊢ (𝐴 ↾ (𝐵 ∪ 𝐶)) = (𝐴 ∩ ((𝐵 ∪ 𝐶) × V)) | |
6 | df-res 4551 | . . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V)) | |
7 | df-res 4551 | . . 3 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V)) | |
8 | 6, 7 | uneq12i 3228 | . 2 ⊢ ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶)) = ((𝐴 ∩ (𝐵 × V)) ∪ (𝐴 ∩ (𝐶 × V))) |
9 | 4, 5, 8 | 3eqtr4i 2170 | 1 ⊢ (𝐴 ↾ (𝐵 ∪ 𝐶)) = ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 Vcvv 2686 ∪ cun 3069 ∩ cin 3070 × cxp 4537 ↾ cres 4541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-in 3077 df-opab 3990 df-xp 4545 df-res 4551 |
This theorem is referenced by: imaundi 4951 relresfld 5068 relcoi1 5070 resasplitss 5302 fnsnsplitss 5619 fnsnsplitdc 6401 fnfi 6825 fseq1p1m1 9874 resunimafz0 10574 |
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