Theorem dmsnop 6165

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 6162	. 2 ⊢ (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = {𝐴})
3	1, 2	ax-mp 5	1 ⊢ dom {⟨𝐴, 𝐵⟩} = {𝐴}

Syntax hints:   = wceq 1540   ∈ wcel 2109  Vcvv 3436  {csn 4577  ⟨cop 4583  dom cdm 5619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-dm 5629

This theorem is referenced by:  dmtpop  6167  dmsnsnsn  6169  op1sta  6174  snres0  6246  funtp  6539  funopdmsn  7084  frrlem14  8232  tfrlem10  8309  ac6sfi  9173  dcomex  10341  axdc3lem4  10347  cnfldfunALT  21276  cnfldfunALTOLD  21289  noextend  27576  nosupbday  27615  nosupbnd1  27624  nosupbnd2  27626  noinfbday  27630  noinfbnd1  27639  noinfbnd2  27641  bnj1416  35006  bnj1421  35009  fineqvac  35072  subfacp1lem2a  35153  subfacp1lem5  35157