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Theorem dmuni 4604
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 1595 . . . . 5 (∃𝑧𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴) ↔ ∃𝑥𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴))
2 ancom 262 . . . . . . 7 ((∃𝑧𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴) ↔ (𝑥𝐴 ∧ ∃𝑧𝑦, 𝑧⟩ ∈ 𝑥))
3 19.41v 1825 . . . . . . 7 (∃𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴) ↔ (∃𝑧𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴))
4 vex 2615 . . . . . . . . 9 𝑦 ∈ V
54eldm2 4592 . . . . . . . 8 (𝑦 ∈ dom 𝑥 ↔ ∃𝑧𝑦, 𝑧⟩ ∈ 𝑥)
65anbi2i 445 . . . . . . 7 ((𝑥𝐴𝑦 ∈ dom 𝑥) ↔ (𝑥𝐴 ∧ ∃𝑧𝑦, 𝑧⟩ ∈ 𝑥))
72, 3, 63bitr4i 210 . . . . . 6 (∃𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴) ↔ (𝑥𝐴𝑦 ∈ dom 𝑥))
87exbii 1537 . . . . 5 (∃𝑥𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴) ↔ ∃𝑥(𝑥𝐴𝑦 ∈ dom 𝑥))
91, 8bitri 182 . . . 4 (∃𝑧𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴) ↔ ∃𝑥(𝑥𝐴𝑦 ∈ dom 𝑥))
10 eluni 3630 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ 𝐴 ↔ ∃𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴))
1110exbii 1537 . . . 4 (∃𝑧𝑦, 𝑧⟩ ∈ 𝐴 ↔ ∃𝑧𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴))
12 df-rex 2359 . . . 4 (∃𝑥𝐴 𝑦 ∈ dom 𝑥 ↔ ∃𝑥(𝑥𝐴𝑦 ∈ dom 𝑥))
139, 11, 123bitr4i 210 . . 3 (∃𝑧𝑦, 𝑧⟩ ∈ 𝐴 ↔ ∃𝑥𝐴 𝑦 ∈ dom 𝑥)
144eldm2 4592 . . 3 (𝑦 ∈ dom 𝐴 ↔ ∃𝑧𝑦, 𝑧⟩ ∈ 𝐴)
15 eliun 3708 . . 3 (𝑦 𝑥𝐴 dom 𝑥 ↔ ∃𝑥𝐴 𝑦 ∈ dom 𝑥)
1613, 14, 153bitr4i 210 . 2 (𝑦 ∈ dom 𝐴𝑦 𝑥𝐴 dom 𝑥)
1716eqriv 2080 1 dom 𝐴 = 𝑥𝐴 dom 𝑥
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1285  wex 1422  wcel 1434  wrex 2354  cop 3425   cuni 3627   ciun 3704  dom cdm 4401
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-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-un 2988  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-iun 3706  df-br 3812  df-dm 4411
This theorem is referenced by:  tfrlem8  6015  tfrlemi14d  6030  tfr1onlemres  6046  tfrcllemres  6059
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