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Theorem coiun 5382
 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 5371 . 2
2 reliun 4998 . . 3
3 relco 5371 . . . 4
43a1i 11 . . 3
52, 4mprgbir 2778 . 2
6 eliun 4099 . . . . . . . 8
7 df-br 4216 . . . . . . . 8
8 df-br 4216 . . . . . . . . 9
98rexbii 2732 . . . . . . . 8
106, 7, 93bitr4i 270 . . . . . . 7
1110anbi1i 678 . . . . . 6
12 r19.41v 2863 . . . . . 6
1311, 12bitr4i 245 . . . . 5
1413exbii 1593 . . . 4
15 rexcom4 2977 . . . 4
1614, 15bitr4i 245 . . 3
17 vex 2961 . . . 4
18 vex 2961 . . . 4
1917, 18opelco 5047 . . 3
20 eliun 4099 . . . 4
2117, 18opelco 5047 . . . . 5
2221rexbii 2732 . . . 4
2320, 22bitri 242 . . 3
2416, 19, 233bitr4i 270 . 2
251, 5, 24eqrelriiv 4973 1
 Colors of variables: wff set class Syntax hints:   wa 360  wex 1551   wceq 1653   wcel 1726  wrex 2708  cop 3819  ciun 4095   class class class wbr 4215   ccom 4885   wrel 4886 This theorem is referenced by:  fparlem3  6451  fparlem4  6452 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 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406 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 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-iun 4097  df-br 4216  df-opab 4270  df-xp 4887  df-rel 4888  df-co 4890
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