Theorem dmsnop 6219

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 6216	. 2 ⊢ (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = {𝐴})
3	1, 2	ax-mp 5	1 ⊢ dom {⟨𝐴, 𝐵⟩} = {𝐴}

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2146  Vcvv 3457  {csn 4591  ⟨cop 4597  dom cdm 5663

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-dm 5673

This theorem is used by:  dmtpop  6221  dmsnsnsn  6223  op1sta  6228  snres0  6303  funtp  6597  funopdmsn  7153  frrlem14  8302  tfrlem10  8380  ac6sfi  9251  dcomex  10446  axdc3lem4  10452  cnfldfunALT  21587  noextend  27881  nosupbday  27920  nosupbnd1  27929  nosupbnd2  27931  noinfbday  27935  noinfbnd1  27944  noinfbnd2  27946  bnj1416  35492  bnj1421  35495  fineqvac  35586  subfacp1lem2a  35709  subfacp1lem5  35713