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Theorem dmuni 5489
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 2191 . . . . 5 (∃𝑧𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴) ↔ ∃𝑥𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴))
2 ancom 465 . . . . . . 7 ((∃𝑧𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴) ↔ (𝑥𝐴 ∧ ∃𝑧𝑦, 𝑧⟩ ∈ 𝑥))
3 19.41v 2026 . . . . . . 7 (∃𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴) ↔ (∃𝑧𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴))
4 vex 3343 . . . . . . . . 9 𝑦 ∈ V
54eldm2 5477 . . . . . . . 8 (𝑦 ∈ dom 𝑥 ↔ ∃𝑧𝑦, 𝑧⟩ ∈ 𝑥)
65anbi2i 732 . . . . . . 7 ((𝑥𝐴𝑦 ∈ dom 𝑥) ↔ (𝑥𝐴 ∧ ∃𝑧𝑦, 𝑧⟩ ∈ 𝑥))
72, 3, 63bitr4i 292 . . . . . 6 (∃𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴) ↔ (𝑥𝐴𝑦 ∈ dom 𝑥))
87exbii 1923 . . . . 5 (∃𝑥𝑧(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴) ↔ ∃𝑥(𝑥𝐴𝑦 ∈ dom 𝑥))
91, 8bitri 264 . . . 4 (∃𝑧𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴) ↔ ∃𝑥(𝑥𝐴𝑦 ∈ dom 𝑥))
10 eluni 4591 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ 𝐴 ↔ ∃𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴))
1110exbii 1923 . . . 4 (∃𝑧𝑦, 𝑧⟩ ∈ 𝐴 ↔ ∃𝑧𝑥(⟨𝑦, 𝑧⟩ ∈ 𝑥𝑥𝐴))
12 df-rex 3056 . . . 4 (∃𝑥𝐴 𝑦 ∈ dom 𝑥 ↔ ∃𝑥(𝑥𝐴𝑦 ∈ dom 𝑥))
139, 11, 123bitr4i 292 . . 3 (∃𝑧𝑦, 𝑧⟩ ∈ 𝐴 ↔ ∃𝑥𝐴 𝑦 ∈ dom 𝑥)
144eldm2 5477 . . 3 (𝑦 ∈ dom 𝐴 ↔ ∃𝑧𝑦, 𝑧⟩ ∈ 𝐴)
15 eliun 4676 . . 3 (𝑦 𝑥𝐴 dom 𝑥 ↔ ∃𝑥𝐴 𝑦 ∈ dom 𝑥)
1613, 14, 153bitr4i 292 . 2 (𝑦 ∈ dom 𝐴𝑦 𝑥𝐴 dom 𝑥)
1716eqriv 2757 1 dom 𝐴 = 𝑥𝐴 dom 𝑥
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1632  wex 1853  wcel 2139  wrex 3051  cop 4327   cuni 4588   ciun 4672  dom cdm 5266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-dm 5276
This theorem is referenced by:  wfrdmss  7591  wfrdmcl  7593  tfrlem8  7650  axdc3lem2  9485  bnj1400  31234  frrlem5d  32114  frrlem5e  32115  frrlem7  32117
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