Theorem dmtpop 5141

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

Step	Hyp	Ref	Expression
1		df-tp 3626	. . . 4 ⊢ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
2	1	dmeqi 4863	. . 3 ⊢ dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = dom ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
3		dmun 4869	. . 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
6	4, 5	dmprop 5140	. . . 4 ⊢ dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶}
7		dmtpop.1	. . . . 5 ⊢ 𝐹 ∈ V
8	7	dmsnop 5139	. . . 4 ⊢ dom {⟨𝐸, 𝐹⟩} = {𝐸}
9	6, 8	uneq12i 3311	. . 3 ⊢ (dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ dom {⟨𝐸, 𝐹⟩}) = ({𝐴, 𝐶} ∪ {𝐸})
10	2, 3, 9	3eqtri 2218	. 2 ⊢ dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = ({𝐴, 𝐶} ∪ {𝐸})
11		df-tp 3626	. 2 ⊢ {𝐴, 𝐶, 𝐸} = ({𝐴, 𝐶} ∪ {𝐸})
12	10, 11	eqtr4i 2217	1 ⊢ dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = {𝐴, 𝐶, 𝐸}

Syntax hints:   = wceq 1364   ∈ wcel 2164  Vcvv 2760   ∪ cun 3151  {csn 3618  {cpr 3619  {ctp 3620  ⟨cop 3621  dom cdm 4659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-tp 3626  df-op 3627  df-br 4030  df-dm 4669

This theorem is referenced by:  fntp  5311