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Theorem dmuni 5661
 Description: The domain of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by NM, 3-Feb-2004.)
Assertion
Ref Expression
dmuni dom 𝐴 = 𝑥𝐴 dom 𝑥
Distinct variable group:   𝑥,𝐴

Proof of Theorem dmuni
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 excom 2131 . . . . 5 (∃𝑧𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴) ↔ ∃𝑥𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴))
2 ancom 461 . . . . . . 7 ((∃𝑧𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴) ↔ (𝑥𝐴 ∧ ∃𝑧𝑦, 𝑧⟩ ∈ 𝑥))
3 19.41v 1925 . . . . . . 7 (∃𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴) ↔ (∃𝑧𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴))
4 vex 3435 . . . . . . . . 9 𝑦 ∈ V
54eldm2 5648 . . . . . . . 8 (𝑦 ∈ dom 𝑥 ↔ ∃𝑧𝑦, 𝑧⟩ ∈ 𝑥)
65anbi2i 622 . . . . . . 7 ((𝑥𝐴𝑦 ∈ dom 𝑥) ↔ (𝑥𝐴 ∧ ∃𝑧𝑦, 𝑧⟩ ∈ 𝑥))
72, 3, 63bitr4i 304 . . . . . 6 (∃𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴) ↔ (𝑥𝐴𝑦 ∈ dom 𝑥))
87exbii 1827 . . . . 5 (∃𝑥𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴) ↔ ∃𝑥(𝑥𝐴𝑦 ∈ dom 𝑥))
91, 8bitri 276 . . . 4 (∃𝑧𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴) ↔ ∃𝑥(𝑥𝐴𝑦 ∈ dom 𝑥))
10 eluni 4742 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ 𝐴 ↔ ∃𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴))
1110exbii 1827 . . . 4 (∃𝑧𝑦, 𝑧⟩ ∈ 𝐴 ↔ ∃𝑧𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴))
12 df-rex 3109 . . . 4 (∃𝑥𝐴 𝑦 ∈ dom 𝑥 ↔ ∃𝑥(𝑥𝐴𝑦 ∈ dom 𝑥))
139, 11, 123bitr4i 304 . . 3 (∃𝑧𝑦, 𝑧⟩ ∈ 𝐴 ↔ ∃𝑥𝐴 𝑦 ∈ dom 𝑥)
144eldm2 5648 . . 3 (𝑦 ∈ dom 𝐴 ↔ ∃𝑧𝑦, 𝑧⟩ ∈ 𝐴)
15 eliun 4823 . . 3 (𝑦 𝑥𝐴 dom 𝑥 ↔ ∃𝑥𝐴 𝑦 ∈ dom 𝑥)
1613, 14, 153bitr4i 304 . 2 (𝑦 ∈ dom 𝐴𝑦 𝑥𝐴 dom 𝑥)
1716eqriv 2790 1 dom 𝐴 = 𝑥𝐴 dom 𝑥
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 396   = wceq 1520  ∃wex 1759   ∈ wcel 2079  ∃wrex 3104  ⟨cop 4472  ∪ cuni 4739  ∪ ciun 4819  dom cdm 5435 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-ext 2767 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-iun 4821  df-br 4957  df-dm 5445 This theorem is referenced by:  wfrdmss  7804  wfrdmcl  7806  tfrlem8  7863  axdc3lem2  9708  bnj1400  31680  frrlem7  32683
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