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Theorem dmprop 6239
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
StepHypRef Expression
1 dmsnop.1 . 2 𝐵 ∈ V
2 dmprop.1 . 2 𝐷 ∈ V
3 dmpropg 6237 . 2 ((𝐵 ∈ V ∧ 𝐷 ∈ V) → dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶})
41, 2, 3mp2an 692 1 dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  Vcvv 3478  {cpr 4633  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-op 4638  df-br 5149  df-dm 5699
This theorem is referenced by:  dmtpop  6240  funtp  6625  fpr  7174  fnprb  7228  hashfun  14473  umgr2v2evd2  29560  ex-dm  30468
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