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Theorem dmiun 4718
Description: The domain of an indexed union. (Contributed by Mario Carneiro, 26-Apr-2016.)
Assertion
Ref Expression
dmiun dom 𝑥𝐴 𝐵 = 𝑥𝐴 dom 𝐵

Proof of Theorem dmiun
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexcom4 2683 . . . 4 (∃𝑥𝐴𝑧𝑦, 𝑧⟩ ∈ 𝐵 ↔ ∃𝑧𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
2 vex 2663 . . . . . 6 𝑦 ∈ V
32eldm2 4707 . . . . 5 (𝑦 ∈ dom 𝐵 ↔ ∃𝑧𝑦, 𝑧⟩ ∈ 𝐵)
43rexbii 2419 . . . 4 (∃𝑥𝐴 𝑦 ∈ dom 𝐵 ↔ ∃𝑥𝐴𝑧𝑦, 𝑧⟩ ∈ 𝐵)
5 eliun 3787 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
65exbii 1569 . . . 4 (∃𝑧𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑧𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
71, 4, 63bitr4ri 212 . . 3 (∃𝑧𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦 ∈ dom 𝐵)
82eldm2 4707 . . 3 (𝑦 ∈ dom 𝑥𝐴 𝐵 ↔ ∃𝑧𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵)
9 eliun 3787 . . 3 (𝑦 𝑥𝐴 dom 𝐵 ↔ ∃𝑥𝐴 𝑦 ∈ dom 𝐵)
107, 8, 93bitr4i 211 . 2 (𝑦 ∈ dom 𝑥𝐴 𝐵𝑦 𝑥𝐴 dom 𝐵)
1110eqriv 2114 1 dom 𝑥𝐴 𝐵 = 𝑥𝐴 dom 𝐵
Colors of variables: wff set class
Syntax hints:   = wceq 1316  wex 1453  wcel 1465  wrex 2394  cop 3500   ciun 3783  dom cdm 4509
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-iun 3785  df-br 3900  df-dm 4519
This theorem is referenced by:  ennnfonelemdm  11860
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