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Theorem dmprop 6052
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 6050 . 2 ((𝐵 ∈ V ∧ 𝐷 ∈ V) → dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶})
41, 2, 3mp2an 691 1 dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wcel 2114  Vcvv 3469  {cpr 4541  cop 4545  dom cdm 5532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-dm 5542
This theorem is referenced by:  dmtpop  6053  funtp  6390  fpr  6898  fnprb  6953  hashfun  13794  umgr2v2evd2  27315  ex-dm  28222
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