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Theorem dmtpop 4824
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 3411 . . . 4 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
21dmeqi 4564 . . 3 dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = dom ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
3 dmun 4570 . . 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 4823 . . . 4 dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶}
7 dmtpop.1 . . . . 5 𝐹 ∈ V
87dmsnop 4822 . . . 4 dom {⟨𝐸, 𝐹⟩} = {𝐸}
96, 8uneq12i 3123 . . 3 (dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ dom {⟨𝐸, 𝐹⟩}) = ({𝐴, 𝐶} ∪ {𝐸})
102, 3, 93eqtri 2080 . 2 dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = ({𝐴, 𝐶} ∪ {𝐸})
11 df-tp 3411 . 2 {𝐴, 𝐶, 𝐸} = ({𝐴, 𝐶} ∪ {𝐸})
1210, 11eqtr4i 2079 1 dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = {𝐴, 𝐶, 𝐸}
Colors of variables: wff set class
Syntax hints:   = wceq 1259  wcel 1409  Vcvv 2574  cun 2943  {csn 3403  {cpr 3404  {ctp 3405  cop 3406  dom cdm 4373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-tp 3411  df-op 3412  df-br 3793  df-dm 4383
This theorem is referenced by:  fntp  4984
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