Theorem dmprop 6120

Description: The domain of an unordered pair of ordered pairs. (Contributed by NM, 13-Sep-2011.)

Hypotheses

Ref	Expression
dmsnop.1	⊢ 𝐵 ∈ V
dmprop.1	⊢ 𝐷 ∈ V

Assertion

Ref	Expression
dmprop	⊢ dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶}

Proof of Theorem dmprop

Step	Hyp	Ref	Expression
1		dmsnop.1	. 2 ⊢ 𝐵 ∈ V
2		dmprop.1	. 2 ⊢ 𝐷 ∈ V
3		dmpropg 6118	. 2 ⊢ ((𝐵 ∈ V ∧ 𝐷 ∈ V) → dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶})
4	1, 2, 3	mp2an 689	1 ⊢ dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶}

Syntax hints:   = wceq 1539   ∈ wcel 2106  Vcvv 3432  {cpr 4563  ⟨cop 4567  dom cdm 5589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-dm 5599

This theorem is referenced by:  dmtpop  6121  funtp  6491  fpr  7026  fnprb  7084  hashfun  14152  umgr2v2evd2  27894  ex-dm  28803