Theorem dmsnop 6218

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 6215	. 2 ⊢ (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = {𝐴})
3	1, 2	ax-mp 5	1 ⊢ dom {⟨𝐴, 𝐵⟩} = {𝐴}

Syntax hints:   = wceq 1567   ∈ wcel 2149  Vcvv 3463  {csn 4594  ⟨cop 4600  dom cdm 5662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-dm 5672

This theorem is referenced by:  dmtpop  6220  dmsnsnsn  6222  op1sta  6227  snres0  6300  funtp  6594  funopdmsn  7148  frrlem14  8295  tfrlem10  8373  ac6sfi  9243  dcomex  10430  axdc3lem4  10436  cnfldfunALT  21505  noextend  27795  nosupbday  27834  nosupbnd1  27843  nosupbnd2  27845  noinfbday  27849  noinfbnd1  27858  noinfbnd2  27860  bnj1416  35371  bnj1421  35374  fineqvac  35451  subfacp1lem2a  35570  subfacp1lem5  35574