MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmiun Structured version   Visualization version   GIF version

Theorem dmiun 5239
Description: The domain of an indexed union. (Contributed by Mario Carneiro, 26-Apr-2016.)
Assertion
Ref Expression
dmiun dom 𝑥𝐴 𝐵 = 𝑥𝐴 dom 𝐵

Proof of Theorem dmiun
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexcom4 3194 . . . 4 (∃𝑥𝐴𝑧𝑦, 𝑧⟩ ∈ 𝐵 ↔ ∃𝑧𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
2 vex 3172 . . . . . 6 𝑦 ∈ V
32eldm2 5228 . . . . 5 (𝑦 ∈ dom 𝐵 ↔ ∃𝑧𝑦, 𝑧⟩ ∈ 𝐵)
43rexbii 3019 . . . 4 (∃𝑥𝐴 𝑦 ∈ dom 𝐵 ↔ ∃𝑥𝐴𝑧𝑦, 𝑧⟩ ∈ 𝐵)
5 eliun 4451 . . . . 5 (⟨𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
65exbii 1763 . . . 4 (∃𝑧𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑧𝑥𝐴𝑦, 𝑧⟩ ∈ 𝐵)
71, 4, 63bitr4ri 291 . . 3 (∃𝑧𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦 ∈ dom 𝐵)
82eldm2 5228 . . 3 (𝑦 ∈ dom 𝑥𝐴 𝐵 ↔ ∃𝑧𝑦, 𝑧⟩ ∈ 𝑥𝐴 𝐵)
9 eliun 4451 . . 3 (𝑦 𝑥𝐴 dom 𝐵 ↔ ∃𝑥𝐴 𝑦 ∈ dom 𝐵)
107, 8, 93bitr4i 290 . 2 (𝑦 ∈ dom 𝑥𝐴 𝐵𝑦 𝑥𝐴 dom 𝐵)
1110eqriv 2603 1 dom 𝑥𝐴 𝐵 = 𝑥𝐴 dom 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  wex 1694  wcel 1976  wrex 2893  cop 4127   ciun 4446  dom cdm 5025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-iun 4448  df-br 4575  df-dm 5035
This theorem is referenced by:  dprd2d2  18209  esum2d  29285  iunrelexp0  36813
  Copyright terms: Public domain W3C validator