Theorem dmsnop 6175

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 6172	. 2 ⊢ (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = {𝐴})
3	1, 2	ax-mp 5	1 ⊢ dom {⟨𝐴, 𝐵⟩} = {𝐴}

Syntax hints:   = wceq 1542   ∈ wcel 2114  Vcvv 3430  {csn 4568  ⟨cop 4574  dom cdm 5625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-dm 5635

This theorem is referenced by:  dmtpop  6177  dmsnsnsn  6179  op1sta  6184  snres0  6257  funtp  6550  funopdmsn  7098  frrlem14  8243  tfrlem10  8320  ac6sfi  9188  dcomex  10363  axdc3lem4  10369  cnfldfunALT  21362  cnfldfunALTOLD  21375  noextend  27647  nosupbday  27686  nosupbnd1  27695  nosupbnd2  27697  noinfbday  27701  noinfbnd1  27710  noinfbnd2  27712  bnj1416  35200  bnj1421  35203  fineqvac  35279  subfacp1lem2a  35381  subfacp1lem5  35385