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Theorem dmiun 4544
 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 2577 . . . 4 (∃𝑥𝐴𝑧𝑦, 𝑧⟩ ∈ 𝐵 ↔ ∃𝑧𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
2 vex 2560 . . . . . 6 𝑦 ∈ V
32eldm2 4533 . . . . 5 (𝑦 ∈ dom 𝐵 ↔ ∃𝑧𝑦, 𝑧⟩ ∈ 𝐵)
43rexbii 2331 . . . 4 (∃𝑥𝐴 𝑦 ∈ dom 𝐵 ↔ ∃𝑥𝐴𝑧𝑦, 𝑧⟩ ∈ 𝐵)
5 eliun 3661 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
65exbii 1496 . . . 4 (∃𝑧𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑧𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
71, 4, 63bitr4ri 202 . . 3 (∃𝑧𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦 ∈ dom 𝐵)
82eldm2 4533 . . 3 (𝑦 ∈ dom 𝑥𝐴 𝐵 ↔ ∃𝑧𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵)
9 eliun 3661 . . 3 (𝑦 𝑥𝐴 dom 𝐵 ↔ ∃𝑥𝐴 𝑦 ∈ dom 𝐵)
107, 8, 93bitr4i 201 . 2 (𝑦 ∈ dom 𝑥𝐴 𝐵𝑦 𝑥𝐴 dom 𝐵)
1110eqriv 2037 1 dom 𝑥𝐴 𝐵 = 𝑥𝐴 dom 𝐵
 Colors of variables: wff set class Syntax hints:   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∃wrex 2307  ⟨cop 3378  ∪ ciun 3657  dom cdm 4345 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-iun 3659  df-br 3765  df-dm 4355 This theorem is referenced by: (None)
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