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Theorem rniun 5978
 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 3177 . . . 4 (∃𝑥𝐴𝑦𝑦, 𝑧⟩ ∈ 𝐵 ↔ ∃𝑦𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
2 vex 3413 . . . . . 6 𝑧 ∈ V
32elrn2 5732 . . . . 5 (𝑧 ∈ ran 𝐵 ↔ ∃𝑦𝑦, 𝑧⟩ ∈ 𝐵)
43rexbii 3175 . . . 4 (∃𝑥𝐴 𝑧 ∈ ran 𝐵 ↔ ∃𝑥𝐴𝑦𝑦, 𝑧⟩ ∈ 𝐵)
5 eliun 4887 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
65exbii 1849 . . . 4 (∃𝑦𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑦𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
71, 4, 63bitr4ri 307 . . 3 (∃𝑦𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑧 ∈ ran 𝐵)
82elrn2 5732 . . 3 (𝑧 ∈ ran 𝑥𝐴 𝐵 ↔ ∃𝑦𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵)
9 eliun 4887 . . 3 (𝑧 𝑥𝐴 ran 𝐵 ↔ ∃𝑥𝐴 𝑧 ∈ ran 𝐵)
107, 8, 93bitr4i 306 . 2 (𝑧 ∈ ran 𝑥𝐴 𝐵𝑧 𝑥𝐴 ran 𝐵)
1110eqriv 2755 1 ran 𝑥𝐴 𝐵 = 𝑥𝐴 ran 𝐵
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃wrex 3071  ⟨cop 4528  ∪ ciun 4883  ran crn 5525 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-11 2158  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-v 3411  df-dif 3861  df-un 3863  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-iun 4885  df-br 5033  df-opab 5095  df-cnv 5532  df-dm 5534  df-rn 5535 This theorem is referenced by:  rnuni  5979  fiun  7648  f1iun  7649  cnextf  22766  iunrelexp0  40776
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