Theorem dmsnop 6192

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 6189	. 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 3450  {csn 4592  ⟨cop 4598  dom cdm 5641

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-dm 5651

This theorem is referenced by:  dmtpop  6194  dmsnsnsn  6196  op1sta  6201  snres0  6274  funtp  6576  funopdmsn  7125  frrlem14  8281  tfrlem10  8358  ac6sfi  9238  dcomex  10407  axdc3lem4  10413  cnfldfunALT  21286  cnfldfunALTOLD  21299  noextend  27585  nosupbday  27624  nosupbnd1  27633  nosupbnd2  27635  noinfbday  27639  noinfbnd1  27648  noinfbnd2  27650  bnj1416  35036  bnj1421  35039  fineqvac  35094  subfacp1lem2a  35174  subfacp1lem5  35178