Theorem dmsnop 6247

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 6244	. 2 ⊢ (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = {𝐴})
3	1, 2	ax-mp 5	1 ⊢ dom {⟨𝐴, 𝐵⟩} = {𝐴}

Syntax hints:   = wceq 1537   ∈ wcel 2108  Vcvv 3488  {csn 4648  ⟨cop 4654  dom cdm 5700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-dm 5710

This theorem is referenced by:  dmtpop  6249  dmsnsnsn  6251  op1sta  6256  snres0  6329  funtp  6635  funopdmsn  7184  frrlem14  8340  wfrlem13OLD  8377  wfrlem16OLD  8380  tfrlem10  8443  ac6sfi  9348  dcomex  10516  axdc3lem4  10522  cnfldfunALT  21402  cnfldfunALTOLD  21415  cnfldfunALTOLDOLD  21416  noextend  27729  nosupbday  27768  nosupbnd1  27777  nosupbnd2  27779  noinfbday  27783  noinfbnd1  27792  noinfbnd2  27794  bnj1416  35015  bnj1421  35018  fineqvac  35073  subfacp1lem2a  35148  subfacp1lem5  35152