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Theorem rniun 4993
 Description: The range of an indexed union. (Contributed by Mario Carneiro, 29-May-2015.)
Assertion
Ref Expression
rniun ran 𝑥𝐴 𝐵 = 𝑥𝐴 ran 𝐵

Proof of Theorem rniun
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexcom4 2735 . . . 4 (∃𝑥𝐴𝑦𝑦, 𝑧⟩ ∈ 𝐵 ↔ ∃𝑦𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
2 vex 2715 . . . . . 6 𝑧 ∈ V
32elrn2 4825 . . . . 5 (𝑧 ∈ ran 𝐵 ↔ ∃𝑦𝑦, 𝑧⟩ ∈ 𝐵)
43rexbii 2464 . . . 4 (∃𝑥𝐴 𝑧 ∈ ran 𝐵 ↔ ∃𝑥𝐴𝑦𝑦, 𝑧⟩ ∈ 𝐵)
5 eliun 3853 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
65exbii 1585 . . . 4 (∃𝑦𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑦𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
71, 4, 63bitr4ri 212 . . 3 (∃𝑦𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑧 ∈ ran 𝐵)
82elrn2 4825 . . 3 (𝑧 ∈ ran 𝑥𝐴 𝐵 ↔ ∃𝑦𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵)
9 eliun 3853 . . 3 (𝑧 𝑥𝐴 ran 𝐵 ↔ ∃𝑥𝐴 𝑧 ∈ ran 𝐵)
107, 8, 93bitr4i 211 . 2 (𝑧 ∈ ran 𝑥𝐴 𝐵𝑧 𝑥𝐴 ran 𝐵)
1110eqriv 2154 1 ran 𝑥𝐴 𝐵 = 𝑥𝐴 ran 𝐵
 Colors of variables: wff set class Syntax hints:   = wceq 1335  ∃wex 1472   ∈ wcel 2128  ∃wrex 2436  ⟨cop 3563  ∪ ciun 3849  ran crn 4584 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-iun 3851  df-br 3966  df-opab 4026  df-cnv 4591  df-dm 4593  df-rn 4594 This theorem is referenced by:  rnuni  4994  fun11iun  5432  ennnfonelemrn  12120
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