MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  coiun Unicode version

Theorem coiun 5425
Description: Composition with an indexed union. (Contributed by NM, 21-Dec-2008.)
Assertion
Ref Expression
coiun
Distinct variable group:   ,
Allowed substitution hints:   ( )   ( )

Proof of Theorem coiun
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relco 5414 . 2
2 reliun 5037 . . 3
3 relco 5414 . . . 4
43a1i 11 . . 3
52, 4mprgbir 2783 . 2
6 eliun 4126 . . . . . . . 8
7 df-br 4244 . . . . . . . 8
8 df-br 4244 . . . . . . . . 9
98rexbii 2737 . . . . . . . 8
106, 7, 93bitr4i 270 . . . . . . 7
1110anbi1i 678 . . . . . 6
12 r19.41v 2868 . . . . . 6
1311, 12bitr4i 245 . . . . 5
1413exbii 1593 . . . 4
15 rexcom4 2984 . . . 4
1614, 15bitr4i 245 . . 3
17 vex 2968 . . . 4
18 vex 2968 . . . 4
1917, 18opelco 5086 . . 3
20 eliun 4126 . . . 4
2117, 18opelco 5086 . . . . 5
2221rexbii 2737 . . . 4
2320, 22bitri 242 . . 3
2416, 19, 233bitr4i 270 . 2
251, 5, 24eqrelriiv 5012 1
Colors of variables: wff set class
Syntax hints:  /\wa 360  E.wex 1551  =wceq 1654  e.wcel 1728  E.wrex 2713  <.cop 3844  U_ciun 4122   class class class wbr 4243  o.ccom 4923  Relwrel 4924
This theorem is referenced by:  fparlem3  6498  fparlem4  6499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4364  ax-nul 4372  ax-pr 4442
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3766  df-sn 3847  df-pr 3848  df-op 3850  df-iun 4124  df-br 4244  df-opab 4302  df-xp 4925  df-rel 4926  df-co 4928
  Copyright terms: Public domain W3C validator