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Theorem dmsnop 4891
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 4889 . 2 (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = {𝐴})
31, 2ax-mp 7 1 dom {⟨𝐴, 𝐵⟩} = {𝐴}
Colors of variables: wff set class
Syntax hints:   = wceq 1289  wcel 1438  Vcvv 2619  {csn 3441  cop 3444  dom cdm 4428
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-dm 4438
This theorem is referenced by:  dmtpop  4893  dmsnsnsng  4895  op1sta  4899  funtp  5053  ac6sfi  6594
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