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Mirrors > Home > ILE Home > Th. List > resundir | GIF version |
Description: Distributive law for restriction over union. (Contributed by NM, 23-Sep-2004.) |
Ref | Expression |
---|---|
resundir | ⊢ ((𝐴 ∪ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | indir 3376 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐶 × V)) ∪ (𝐵 ∩ (𝐶 × V))) | |
2 | df-res 4623 | . 2 ⊢ ((𝐴 ∪ 𝐵) ↾ 𝐶) = ((𝐴 ∪ 𝐵) ∩ (𝐶 × V)) | |
3 | df-res 4623 | . . 3 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V)) | |
4 | df-res 4623 | . . 3 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V)) | |
5 | 3, 4 | uneq12i 3279 | . 2 ⊢ ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶)) = ((𝐴 ∩ (𝐶 × V)) ∪ (𝐵 ∩ (𝐶 × V))) |
6 | 1, 2, 5 | 3eqtr4i 2201 | 1 ⊢ ((𝐴 ∪ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 Vcvv 2730 ∪ cun 3119 ∩ cin 3120 × cxp 4609 ↾ cres 4613 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-un 3125 df-in 3127 df-res 4623 |
This theorem is referenced by: imaundir 5024 fvunsng 5690 fvsnun1 5693 fvsnun2 5694 fsnunfv 5697 fsnunres 5698 fseq1p1m1 10050 setsresg 12454 setscom 12456 setsslid 12466 |
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