Theorem dmsnop 6163

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 6160	. 2 ⊢ (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = {𝐴})
3	1, 2	ax-mp 5	1 ⊢ dom {⟨𝐴, 𝐵⟩} = {𝐴}

Syntax hints:   = wceq 1541   ∈ wcel 2111  Vcvv 3436  {csn 4573  ⟨cop 4579  dom cdm 5614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-dm 5624

This theorem is referenced by:  dmtpop  6165  dmsnsnsn  6167  op1sta  6172  snres0  6245  funtp  6538  funopdmsn  7083  frrlem14  8229  tfrlem10  8306  ac6sfi  9168  dcomex  10338  axdc3lem4  10344  cnfldfunALT  21306  cnfldfunALTOLD  21319  noextend  27605  nosupbday  27644  nosupbnd1  27653  nosupbnd2  27655  noinfbday  27659  noinfbnd1  27668  noinfbnd2  27670  bnj1416  35051  bnj1421  35054  fineqvac  35139  subfacp1lem2a  35224  subfacp1lem5  35228