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Theorem rniun 4764
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 2623 . . . 4 (∃𝑥𝐴𝑦𝑦, 𝑧⟩ ∈ 𝐵 ↔ ∃𝑦𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
2 vex 2605 . . . . . 6 𝑧 ∈ V
32elrn2 4604 . . . . 5 (𝑧 ∈ ran 𝐵 ↔ ∃𝑦𝑦, 𝑧⟩ ∈ 𝐵)
43rexbii 2374 . . . 4 (∃𝑥𝐴 𝑧 ∈ ran 𝐵 ↔ ∃𝑥𝐴𝑦𝑦, 𝑧⟩ ∈ 𝐵)
5 eliun 3690 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
65exbii 1537 . . . 4 (∃𝑦𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑦𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
71, 4, 63bitr4ri 211 . . 3 (∃𝑦𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑧 ∈ ran 𝐵)
82elrn2 4604 . . 3 (𝑧 ∈ ran 𝑥𝐴 𝐵 ↔ ∃𝑦𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵)
9 eliun 3690 . . 3 (𝑧 𝑥𝐴 ran 𝐵 ↔ ∃𝑥𝐴 𝑧 ∈ ran 𝐵)
107, 8, 93bitr4i 210 . 2 (𝑧 ∈ ran 𝑥𝐴 𝐵𝑧 𝑥𝐴 ran 𝐵)
1110eqriv 2079 1 ran 𝑥𝐴 𝐵 = 𝑥𝐴 ran 𝐵
Colors of variables: wff set class
Syntax hints:   = wceq 1285  wex 1422  wcel 1434  wrex 2350  cop 3409   ciun 3686  ran crn 4372
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-iun 3688  df-br 3794  df-opab 3848  df-cnv 4379  df-dm 4381  df-rn 4382
This theorem is referenced by:  rnuni  4765  fun11iun  5178
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