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Mirrors > Home > MPE Home > Th. List > resundi | Structured version Visualization version 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 5585 | . . . 4 ⊢ ((𝐵 ∪ 𝐶) × V) = ((𝐵 × V) ∪ (𝐶 × V)) | |
2 | 1 | ineq2i 4136 | . . 3 ⊢ (𝐴 ∩ ((𝐵 ∪ 𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∪ (𝐶 × V))) |
3 | indi 4200 | . . 3 ⊢ (𝐴 ∩ ((𝐵 × V) ∪ (𝐶 × V))) = ((𝐴 ∩ (𝐵 × V)) ∪ (𝐴 ∩ (𝐶 × V))) | |
4 | 2, 3 | eqtri 2821 | . 2 ⊢ (𝐴 ∩ ((𝐵 ∪ 𝐶) × V)) = ((𝐴 ∩ (𝐵 × V)) ∪ (𝐴 ∩ (𝐶 × V))) |
5 | df-res 5531 | . 2 ⊢ (𝐴 ↾ (𝐵 ∪ 𝐶)) = (𝐴 ∩ ((𝐵 ∪ 𝐶) × V)) | |
6 | df-res 5531 | . . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V)) | |
7 | df-res 5531 | . . 3 ⊢ (𝐴 ↾ 𝐶) = (𝐴 ∩ (𝐶 × V)) | |
8 | 6, 7 | uneq12i 4088 | . 2 ⊢ ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶)) = ((𝐴 ∩ (𝐵 × V)) ∪ (𝐴 ∩ (𝐶 × V))) |
9 | 4, 5, 8 | 3eqtr4i 2831 | 1 ⊢ (𝐴 ↾ (𝐵 ∪ 𝐶)) = ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 Vcvv 3441 ∪ cun 3879 ∩ cin 3880 × cxp 5517 ↾ cres 5521 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-opab 5093 df-xp 5525 df-res 5531 |
This theorem is referenced by: imaundi 5975 relresfld 6095 resasplit 6522 fresaunres2 6524 residpr 6882 fnsnsplit 6923 tfrlem16 8012 mapunen 8670 fnfi 8780 fseq1p1m1 12976 resunimafz0 13799 gsum2dlem2 19084 dprd2da 19157 evlseu 20755 ptuncnv 22412 mbfres2 24249 reldisjun 30366 ffsrn 30491 resf1o 30492 symgcom 30777 tocyc01 30810 cvmliftlem10 32654 eqfunresadj 33117 nosupbnd2lem1 33328 poimirlem9 35066 eldioph4b 39752 pwssplit4 40033 undmrnresiss 40304 relexp0a 40417 rnresun 41804 |
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