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Mirrors > Home > ILE Home > Th. List > resundi | Unicode version |
Description: Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.) |
Ref | Expression |
---|---|
resundi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpundir 4591 | . . . 4 | |
2 | 1 | ineq2i 3269 | . . 3 |
3 | indi 3318 | . . 3 | |
4 | 2, 3 | eqtri 2158 | . 2 |
5 | df-res 4546 | . 2 | |
6 | df-res 4546 | . . 3 | |
7 | df-res 4546 | . . 3 | |
8 | 6, 7 | uneq12i 3223 | . 2 |
9 | 4, 5, 8 | 3eqtr4i 2168 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1331 cvv 2681 cun 3064 cin 3065 cxp 4532 cres 4536 |
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 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-un 3070 df-in 3072 df-opab 3985 df-xp 4540 df-res 4546 |
This theorem is referenced by: imaundi 4946 relresfld 5063 relcoi1 5065 resasplitss 5297 fnsnsplitss 5612 fnsnsplitdc 6394 fnfi 6818 fseq1p1m1 9867 resunimafz0 10567 |
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