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Theorem dmtpop 6174
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 4595 . . . 4 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
21dmeqi 5864 . . 3 dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = dom ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
3 dmun 5870 . . 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 6173 . . . 4 dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶}
7 dmtpop.1 . . . . 5 𝐹 ∈ V
87dmsnop 6172 . . . 4 dom {⟨𝐸, 𝐹⟩} = {𝐸}
96, 8uneq12i 4125 . . 3 (dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ dom {⟨𝐸, 𝐹⟩}) = ({𝐴, 𝐶} ∪ {𝐸})
102, 3, 93eqtri 2765 . 2 dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = ({𝐴, 𝐶} ∪ {𝐸})
11 df-tp 4595 . 2 {𝐴, 𝐶, 𝐸} = ({𝐴, 𝐶} ∪ {𝐸})
1210, 11eqtr4i 2764 1 dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = {𝐴, 𝐶, 𝐸}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  Vcvv 3447  cun 3912  {csn 4590  {cpr 4592  {ctp 4594  cop 4596  dom cdm 5637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-br 5110  df-dm 5647
This theorem is referenced by:  fntp  6566  fntpb  7163  cnfldfunALT  20832  cnfldfunALTOLD  20833
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