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Theorem dmuni 5925
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 2162 . . . . 5 (∃𝑧𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴) ↔ ∃𝑥𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴))
2 ancom 460 . . . . . . 7 ((∃𝑧𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴) ↔ (𝑥𝐴 ∧ ∃𝑧𝑦, 𝑧⟩ ∈ 𝑥))
3 19.41v 1949 . . . . . . 7 (∃𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴) ↔ (∃𝑧𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴))
4 vex 3484 . . . . . . . . 9 𝑦 ∈ V
54eldm2 5912 . . . . . . . 8 (𝑦 ∈ dom 𝑥 ↔ ∃𝑧𝑦, 𝑧⟩ ∈ 𝑥)
65anbi2i 623 . . . . . . 7 ((𝑥𝐴𝑦 ∈ dom 𝑥) ↔ (𝑥𝐴 ∧ ∃𝑧𝑦, 𝑧⟩ ∈ 𝑥))
72, 3, 63bitr4i 303 . . . . . 6 (∃𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴) ↔ (𝑥𝐴𝑦 ∈ dom 𝑥))
87exbii 1848 . . . . 5 (∃𝑥𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴) ↔ ∃𝑥(𝑥𝐴𝑦 ∈ dom 𝑥))
91, 8bitri 275 . . . 4 (∃𝑧𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴) ↔ ∃𝑥(𝑥𝐴𝑦 ∈ dom 𝑥))
10 eluni 4910 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ 𝐴 ↔ ∃𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴))
1110exbii 1848 . . . 4 (∃𝑧𝑦, 𝑧⟩ ∈ 𝐴 ↔ ∃𝑧𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴))
12 df-rex 3071 . . . 4 (∃𝑥𝐴 𝑦 ∈ dom 𝑥 ↔ ∃𝑥(𝑥𝐴𝑦 ∈ dom 𝑥))
139, 11, 123bitr4i 303 . . 3 (∃𝑧𝑦, 𝑧⟩ ∈ 𝐴 ↔ ∃𝑥𝐴 𝑦 ∈ dom 𝑥)
144eldm2 5912 . . 3 (𝑦 ∈ dom 𝐴 ↔ ∃𝑧𝑦, 𝑧⟩ ∈ 𝐴)
15 eliun 4995 . . 3 (𝑦 𝑥𝐴 dom 𝑥 ↔ ∃𝑥𝐴 𝑦 ∈ dom 𝑥)
1613, 14, 153bitr4i 303 . 2 (𝑦 ∈ dom 𝐴𝑦 𝑥𝐴 dom 𝑥)
1716eqriv 2734 1 dom 𝐴 = 𝑥𝐴 dom 𝑥
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wex 1779  wcel 2108  wrex 3070  cop 4632   cuni 4907   ciun 4991  dom cdm 5685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-dm 5695
This theorem is referenced by:  frrlem7  8317  wfrdmssOLD  8355  wfrdmclOLD  8357  tfrlem8  8424  axdc3lem2  10491  bnj1400  34849
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