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Theorem rniun 5034
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 2760 . . . 4 (∃𝑥𝐴𝑦𝑦, 𝑧⟩ ∈ 𝐵 ↔ ∃𝑦𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
2 vex 2740 . . . . . 6 𝑧 ∈ V
32elrn2 4864 . . . . 5 (𝑧 ∈ ran 𝐵 ↔ ∃𝑦𝑦, 𝑧⟩ ∈ 𝐵)
43rexbii 2484 . . . 4 (∃𝑥𝐴 𝑧 ∈ ran 𝐵 ↔ ∃𝑥𝐴𝑦𝑦, 𝑧⟩ ∈ 𝐵)
5 eliun 3888 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
65exbii 1605 . . . 4 (∃𝑦𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑦𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
71, 4, 63bitr4ri 213 . . 3 (∃𝑦𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑧 ∈ ran 𝐵)
82elrn2 4864 . . 3 (𝑧 ∈ ran 𝑥𝐴 𝐵 ↔ ∃𝑦𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵)
9 eliun 3888 . . 3 (𝑧 𝑥𝐴 ran 𝐵 ↔ ∃𝑥𝐴 𝑧 ∈ ran 𝐵)
107, 8, 93bitr4i 212 . 2 (𝑧 ∈ ran 𝑥𝐴 𝐵𝑧 𝑥𝐴 ran 𝐵)
1110eqriv 2174 1 ran 𝑥𝐴 𝐵 = 𝑥𝐴 ran 𝐵
Colors of variables: wff set class
Syntax hints:   = wceq 1353  wex 1492  wcel 2148  wrex 2456  cop 3594   ciun 3884  ran crn 4623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-iun 3886  df-br 4001  df-opab 4062  df-cnv 4630  df-dm 4632  df-rn 4633
This theorem is referenced by:  rnuni  5035  fun11iun  5477  ennnfonelemrn  12390
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