Theorem dmtpop 6238

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 4593	. . . 4 ⊢ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
2	1	dmeqi 5898	. . 3 ⊢ dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = dom ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
3		dmun 5904	. . 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 6237	. . . 4 ⊢ dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶}
7		dmtpop.1	. . . . 5 ⊢ 𝐹 ∈ V
8	7	dmsnop 6236	. . . 4 ⊢ dom {⟨𝐸, 𝐹⟩} = {𝐸}
9	6, 8	uneq12i 4168	. . 3 ⊢ (dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ dom {⟨𝐸, 𝐹⟩}) = ({𝐴, 𝐶} ∪ {𝐸})
10	2, 3, 9	3eqtri 2790	. 2 ⊢ dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = ({𝐴, 𝐶} ∪ {𝐸})
11		df-tp 4593	. 2 ⊢ {𝐴, 𝐶, 𝐸} = ({𝐴, 𝐶} ∪ {𝐸})
12	10, 11	eqtr4i 2789	1 ⊢ dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = {𝐴, 𝐶, 𝐸}

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ∪ cun 4152  {csn 4588  {cpr 4590  {ctp 4592  ⟨cop 4840  dom cdm 5658

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-br 5109  df-dm 5668

This theorem is used by:  fntp  6631  fvtp0  7237  fntpb  7246  degenmgmopdm  19184  degenmgm2opdm  19188  cnfldfunALT  21745