Theorem dmsnop 6216

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 6213	. 2 ⊢ (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = {𝐴})
3	1, 2	ax-mp 5	1 ⊢ dom {⟨𝐴, 𝐵⟩} = {𝐴}

Syntax hints:   = wceq 1542   ∈ wcel 2107  Vcvv 3475  {csn 4629  ⟨cop 4635  dom cdm 5677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-dm 5687

This theorem is referenced by:  dmtpop  6218  dmsnsnsn  6220  op1sta  6225  snres0  6298  funtp  6606  funopdmsn  7148  frrlem14  8284  wfrlem13OLD  8321  wfrlem16OLD  8324  tfrlem10  8387  ac6sfi  9287  dcomex  10442  axdc3lem4  10448  cnfldfunALT  20957  cnfldfunALTOLD  20958  noextend  27169  nosupbday  27208  nosupbnd1  27217  nosupbnd2  27219  noinfbday  27223  noinfbnd1  27232  noinfbnd2  27234  bnj1416  34081  bnj1421  34084  fineqvac  34128  subfacp1lem2a  34202  subfacp1lem5  34206  gg-cnfldfunALT  35229