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Theorem dmtpop 5575
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 4158 . . . 4 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
21dmeqi 5290 . . 3 dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = dom ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
3 dmun 5296 . . 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 5574 . . . 4 dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶}
7 dmtpop.1 . . . . 5 𝐹 ∈ V
87dmsnop 5573 . . . 4 dom {⟨𝐸, 𝐹⟩} = {𝐸}
96, 8uneq12i 3748 . . 3 (dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ dom {⟨𝐸, 𝐹⟩}) = ({𝐴, 𝐶} ∪ {𝐸})
102, 3, 93eqtri 2647 . 2 dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = ({𝐴, 𝐶} ∪ {𝐸})
11 df-tp 4158 . 2 {𝐴, 𝐶, 𝐸} = ({𝐴, 𝐶} ∪ {𝐸})
1210, 11eqtr4i 2646 1 dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = {𝐴, 𝐶, 𝐸}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  Vcvv 3189  cun 3557  {csn 4153  {cpr 4155  {ctp 4157  cop 4159  dom cdm 5079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-br 4619  df-dm 5089
This theorem is referenced by:  fntp  5912  fntpb  6433  cnfldfun  19690
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