Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  dmsnop GIF version

Theorem dmsnop 5007
 Description: The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypothesis
Ref Expression
dmsnop.1 𝐵 ∈ V
Assertion
Ref Expression
dmsnop dom {⟨𝐴, 𝐵⟩} = {𝐴}

Proof of Theorem dmsnop
StepHypRef Expression
1 dmsnop.1 . 2 𝐵 ∈ V
2 dmsnopg 5005 . 2 (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = {𝐴})
31, 2ax-mp 5 1 dom {⟨𝐴, 𝐵⟩} = {𝐴}
 Colors of variables: wff set class Syntax hints:   = wceq 1331   ∈ wcel 1480  Vcvv 2681  {csn 3522  ⟨cop 3525  dom cdm 4534 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-dm 4544 This theorem is referenced by:  dmtpop  5009  dmsnsnsng  5011  op1sta  5015  funtp  5171  ac6sfi  6785
 Copyright terms: Public domain W3C validator