Theorem dmtpop 6165

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 4578	. . . 4 ⊢ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
2	1	dmeqi 5843	. . 3 ⊢ dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = dom ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
3		dmun 5849	. . 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 6164	. . . 4 ⊢ dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶}
7		dmtpop.1	. . . . 5 ⊢ 𝐹 ∈ V
8	7	dmsnop 6163	. . . 4 ⊢ dom {⟨𝐸, 𝐹⟩} = {𝐸}
9	6, 8	uneq12i 4113	. . 3 ⊢ (dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ dom {⟨𝐸, 𝐹⟩}) = ({𝐴, 𝐶} ∪ {𝐸})
10	2, 3, 9	3eqtri 2758	. 2 ⊢ dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = ({𝐴, 𝐶} ∪ {𝐸})
11		df-tp 4578	. 2 ⊢ {𝐴, 𝐶, 𝐸} = ({𝐴, 𝐶} ∪ {𝐸})
12	10, 11	eqtr4i 2757	1 ⊢ dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = {𝐴, 𝐶, 𝐸}

Syntax hints:   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ∪ cun 3895  {csn 4573  {cpr 4575  {ctp 4577  ⟨cop 4579  dom cdm 5614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-br 5090  df-dm 5624

This theorem is referenced by:  fntp  6542  fntpb  7143  cnfldfunALT  21306  cnfldfunALTOLD  21319