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Theorem rniun 5421
Description: The range of an indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
Assertion
Ref Expression
rniun

Proof of Theorem rniun
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexcom4 3129 . . . 4
2 vex 3112 . . . . . 6
32elrn2 5247 . . . . 5
43rexbii 2959 . . . 4
5 eliun 4335 . . . . 5
65exbii 1667 . . . 4
71, 4, 63bitr4ri 278 . . 3
82elrn2 5247 . . 3
9 eliun 4335 . . 3
107, 8, 93bitr4i 277 . 2
1110eqriv 2453 1
Colors of variables: wff setvar class
Syntax hints:  =wceq 1395  E.wex 1612  e.wcel 1818  E.wrex 2808  <.cop 4035  U_ciun 4330  rancrn 5005
This theorem is referenced by:  rnuni  5422  fun11iun  6760  cnextf  20566
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-cnv 5012  df-dm 5014  df-rn 5015
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