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Theorem dmuni 5537
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 2205 . . . . 5 (∃𝑧𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴) ↔ ∃𝑥𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴))
2 ancom 453 . . . . . . 7 ((∃𝑧𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴) ↔ (𝑥𝐴 ∧ ∃𝑧𝑦, 𝑧⟩ ∈ 𝑥))
3 19.41v 2045 . . . . . . 7 (∃𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴) ↔ (∃𝑧𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴))
4 vex 3388 . . . . . . . . 9 𝑦 ∈ V
54eldm2 5525 . . . . . . . 8 (𝑦 ∈ dom 𝑥 ↔ ∃𝑧𝑦, 𝑧⟩ ∈ 𝑥)
65anbi2i 617 . . . . . . 7 ((𝑥𝐴𝑦 ∈ dom 𝑥) ↔ (𝑥𝐴 ∧ ∃𝑧𝑦, 𝑧⟩ ∈ 𝑥))
72, 3, 63bitr4i 295 . . . . . 6 (∃𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴) ↔ (𝑥𝐴𝑦 ∈ dom 𝑥))
87exbii 1944 . . . . 5 (∃𝑥𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴) ↔ ∃𝑥(𝑥𝐴𝑦 ∈ dom 𝑥))
91, 8bitri 267 . . . 4 (∃𝑧𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴) ↔ ∃𝑥(𝑥𝐴𝑦 ∈ dom 𝑥))
10 eluni 4631 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ 𝐴 ↔ ∃𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴))
1110exbii 1944 . . . 4 (∃𝑧𝑦, 𝑧⟩ ∈ 𝐴 ↔ ∃𝑧𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴))
12 df-rex 3095 . . . 4 (∃𝑥𝐴 𝑦 ∈ dom 𝑥 ↔ ∃𝑥(𝑥𝐴𝑦 ∈ dom 𝑥))
139, 11, 123bitr4i 295 . . 3 (∃𝑧𝑦, 𝑧⟩ ∈ 𝐴 ↔ ∃𝑥𝐴 𝑦 ∈ dom 𝑥)
144eldm2 5525 . . 3 (𝑦 ∈ dom 𝐴 ↔ ∃𝑧𝑦, 𝑧⟩ ∈ 𝐴)
15 eliun 4714 . . 3 (𝑦 𝑥𝐴 dom 𝑥 ↔ ∃𝑥𝐴 𝑦 ∈ dom 𝑥)
1613, 14, 153bitr4i 295 . 2 (𝑦 ∈ dom 𝐴𝑦 𝑥𝐴 dom 𝑥)
1716eqriv 2796 1 dom 𝐴 = 𝑥𝐴 dom 𝑥
Colors of variables: wff setvar class
Syntax hints:  wa 385   = wceq 1653  wex 1875  wcel 2157  wrex 3090  cop 4374   cuni 4628   ciun 4710  dom cdm 5312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-dm 5322
This theorem is referenced by:  wfrdmss  7660  wfrdmcl  7662  tfrlem8  7719  axdc3lem2  9561  bnj1400  31423  frrlem5d  32300  frrlem5e  32301  frrlem7  32303
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