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Theorem coiun 5522
Description: Composition with an indexed union. (Contributed by NM, 21-Dec-2008.)
Assertion
Ref Expression
coiun
Distinct variable group:   ,

Proof of Theorem coiun
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relco 5510 . 2
2 reliun 5128 . . 3
3 relco 5510 . . . 4
43a1i 11 . . 3
52, 4mprgbir 2821 . 2
6 eliun 4335 . . . . . . . 8
7 df-br 4453 . . . . . . . 8
8 df-br 4453 . . . . . . . . 9
98rexbii 2959 . . . . . . . 8
106, 7, 93bitr4i 277 . . . . . . 7
1110anbi1i 695 . . . . . 6
12 r19.41v 3009 . . . . . 6
1311, 12bitr4i 252 . . . . 5
1413exbii 1667 . . . 4
15 rexcom4 3129 . . . 4
1614, 15bitr4i 252 . . 3
17 vex 3112 . . . 4
18 vex 3112 . . . 4
1917, 18opelco 5179 . . 3
20 eliun 4335 . . . 4
2117, 18opelco 5179 . . . . 5
2221rexbii 2959 . . . 4
2320, 22bitri 249 . . 3
2416, 19, 233bitr4i 277 . 2
251, 5, 24eqrelriiv 5102 1
Colors of variables: wff setvar class
Syntax hints:  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  E.wrex 2808  <.cop 4035  U_ciun 4330   class class class wbr 4452  o.ccom 5008  Relwrel 5009
This theorem is referenced by:  fparlem3  6902  fparlem4  6903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-iun 4332  df-br 4453  df-opab 4511  df-xp 5010  df-rel 5011  df-co 5013
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