Theorem dmsnop 6214

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 6211	. 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 4591  ⟨cop 4597  dom cdm 5659

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 5258  ax-pr 5402

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 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-dm 5669

This theorem is referenced by:  dmtpop  6216  dmsnsnsn  6218  op1sta  6223  snres0  6296  funtp  6590  funopdmsn  7145  frrlem14  8292  tfrlem10  8370  ac6sfi  9240  dcomex  10427  axdc3lem4  10433  cnfldfunALT  21502  noextend  27792  nosupbday  27831  nosupbnd1  27840  nosupbnd2  27842  noinfbday  27846  noinfbnd1  27855  noinfbnd2  27857  bnj1416  35368  bnj1421  35371  fineqvac  35448  subfacp1lem2a  35567  subfacp1lem5  35571