Theorem dmsnop 5155

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

Step	Hyp	Ref	Expression
1		dmsnop.1	. 2 ⊢ 𝐵 ∈ V
2		dmsnopg 5153	. 2 ⊢ (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = {𝐴})
3	1, 2	ax-mp 5	1 ⊢ dom {⟨𝐴, 𝐵⟩} = {𝐴}

Syntax hints:   = wceq 1372   ∈ wcel 2175  Vcvv 2771  {csn 3632  ⟨cop 3635  dom cdm 4674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-dm 4684

This theorem is referenced by:  dmtpop  5157  dmsnsnsng  5159  op1sta  5163  funtp  5326  funopdmsn  5763  ac6sfi  6994