Theorem dmsnop 6217

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 6214	. 2 ⊢ (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = {𝐴})
3	1, 2	ax-mp 5	1 ⊢ dom {⟨𝐴, 𝐵⟩} = {𝐴}

Syntax hints:   = wceq 1570   ∈ wcel 2143  Vcvv 3455  {csn 4589  ⟨cop 4595  dom cdm 5661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-dm 5671

This theorem is referenced by:  dmtpop  6219  dmsnsnsn  6221  op1sta  6226  snres0  6299  funtp  6593  funopdmsn  7147  frrlem14  8292  tfrlem10  8370  ac6sfi  9240  dcomex  10426  axdc3lem4  10432  cnfldfunALT  21537  noextend  27830  nosupbday  27869  nosupbnd1  27878  nosupbnd2  27880  noinfbday  27884  noinfbnd1  27893  noinfbnd2  27895  bnj1416  35427  bnj1421  35430  fineqvac  35529  subfacp1lem2a  35672  subfacp1lem5  35676