Theorem dmtpop 6240

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 4636	. . . 4 ⊢ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
2	1	dmeqi 5918	. . 3 ⊢ dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = dom ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
3		dmun 5924	. . 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 6239	. . . 4 ⊢ dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶}
7		dmtpop.1	. . . . 5 ⊢ 𝐹 ∈ V
8	7	dmsnop 6238	. . . 4 ⊢ dom {⟨𝐸, 𝐹⟩} = {𝐸}
9	6, 8	uneq12i 4176	. . 3 ⊢ (dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ dom {⟨𝐸, 𝐹⟩}) = ({𝐴, 𝐶} ∪ {𝐸})
10	2, 3, 9	3eqtri 2767	. 2 ⊢ dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = ({𝐴, 𝐶} ∪ {𝐸})
11		df-tp 4636	. 2 ⊢ {𝐴, 𝐶, 𝐸} = ({𝐴, 𝐶} ∪ {𝐸})
12	10, 11	eqtr4i 2766	1 ⊢ dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = {𝐴, 𝐶, 𝐸}

Syntax hints:   = wceq 1537   ∈ wcel 2106  Vcvv 3478   ∪ cun 3961  {csn 4631  {cpr 4633  {ctp 4635  ⟨cop 4637  dom cdm 5689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-br 5149  df-dm 5699

This theorem is referenced by:  fntp  6629  fntpb  7229  cnfldfunALT  21397  cnfldfunALTOLD  21410  cnfldfunALTOLDOLD  21411