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Theorem dmtpop 5529
 Description: The domain of an unordered triple of ordered pairs. (Contributed by NM, 14-Sep-2011.)
Hypotheses
Ref Expression
dmsnop.1 𝐵 ∈ V
dmprop.1 𝐷 ∈ V
dmtpop.1 𝐹 ∈ V
Assertion
Ref Expression
dmtpop dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = {𝐴, 𝐶, 𝐸}

Proof of Theorem dmtpop
StepHypRef Expression
1 df-tp 4130 . . . 4 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
21dmeqi 5247 . . 3 dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = dom ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
3 dmun 5253 . . 3 dom ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩}) = (dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ dom {⟨𝐸, 𝐹⟩})
4 dmsnop.1 . . . . 5 𝐵 ∈ V
5 dmprop.1 . . . . 5 𝐷 ∈ V
64, 5dmprop 5528 . . . 4 dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶}
7 dmtpop.1 . . . . 5 𝐹 ∈ V
87dmsnop 5527 . . . 4 dom {⟨𝐸, 𝐹⟩} = {𝐸}
96, 8uneq12i 3727 . . 3 (dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ dom {⟨𝐸, 𝐹⟩}) = ({𝐴, 𝐶} ∪ {𝐸})
102, 3, 93eqtri 2636 . 2 dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = ({𝐴, 𝐶} ∪ {𝐸})
11 df-tp 4130 . 2 {𝐴, 𝐶, 𝐸} = ({𝐴, 𝐶} ∪ {𝐸})
1210, 11eqtr4i 2635 1 dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = {𝐴, 𝐶, 𝐸}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∪ cun 3538  {csn 4125  {cpr 4127  {ctp 4129  ⟨cop 4131  dom cdm 5038 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-br 4584  df-dm 5048 This theorem is referenced by:  fntp  5863  fntpb  6378
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