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Theorem resundi 4568
Description: Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)
Assertion
Ref Expression
resundi  |`  u.  C  |`  u.  |`  C

Proof of Theorem resundi
StepHypRef Expression
1 xpundir 4340 . . . 4  u.  C  X.  _V  X.  _V  u.  C  X.  _V
21ineq2i 3129 . . 3  i^i  u.  C  X.  _V  i^i  X.  _V  u.  C  X.  _V
3 indi 3178 . . 3  i^i  X.  _V  u.  C  X.  _V  i^i  X.  _V  u.  i^i  C  X.  _V
42, 3eqtri 2057 . 2  i^i  u.  C  X.  _V  i^i  X.  _V  u.  i^i  C  X.  _V
5 df-res 4300 . 2  |`  u.  C  i^i  u.  C  X.  _V
6 df-res 4300 . . 3  |`  i^i  X.  _V
7 df-res 4300 . . 3  |`  C  i^i  C  X.  _V
86, 7uneq12i 3089 . 2  |`  u.  |`  C  i^i  X.  _V  u.  i^i  C  X.  _V
94, 5, 83eqtr4i 2067 1  |`  u.  C  |`  u.  |`  C
Colors of variables: wff set class
Syntax hints:   wceq 1242   _Vcvv 2551    u. cun 2909    i^i cin 2910    X. cxp 4286    |` cres 4290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-opab 3810  df-xp 4294  df-res 4300
This theorem is referenced by:  imaundi  4679  relresfld  4790  relcoi1  4792  resasplitss  5012  fseq1p1m1  8726
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