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Theorem coiun 5371
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 5360 . 2
2 reliun 4987 . . 3
3 relco 5360 . . . 4
43a1i 11 . . 3
52, 4mprgbir 2768 . 2
6 eliun 4089 . . . . . . . 8
7 df-br 4205 . . . . . . . 8
8 df-br 4205 . . . . . . . . 9
98rexbii 2722 . . . . . . . 8
106, 7, 93bitr4i 269 . . . . . . 7
1110anbi1i 677 . . . . . 6
12 r19.41v 2853 . . . . . 6
1311, 12bitr4i 244 . . . . 5
1413exbii 1592 . . . 4
15 rexcom4 2967 . . . 4
1614, 15bitr4i 244 . . 3
17 vex 2951 . . . 4
18 vex 2951 . . . 4
1917, 18opelco 5036 . . 3
20 eliun 4089 . . . 4
2117, 18opelco 5036 . . . . 5
2221rexbii 2722 . . . 4
2320, 22bitri 241 . . 3
2416, 19, 233bitr4i 269 . 2
251, 5, 24eqrelriiv 4962 1
Colors of variables: wff set class
Syntax hints:  /\wa 359  E.wex 1550  =wceq 1652  e.wcel 1725  E.wrex 2698  <.cop 3809  U_ciun 4085   class class class wbr 4204  o.ccom 4874  Relwrel 4875
This theorem is referenced by:  fparlem3  6440  fparlem4  6441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-iun 4087  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-co 4879
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