Theorem dmsnop 6203

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 6200	. 2 ⊢ (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = {𝐴})
3	1, 2	ax-mp 5	1 ⊢ dom {⟨𝐴, 𝐵⟩} = {𝐴}

Syntax hints:   = wceq 1560   ∈ wcel 2142  Vcvv 3454  {csn 4582  ⟨cop 4588  dom cdm 5647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-dm 5657

This theorem is referenced by:  dmtpop  6205  dmsnsnsn  6207  op1sta  6212  snres0  6285  funtp  6578  funopdmsn  7133  frrlem14  8280  tfrlem10  8358  ac6sfi  9228  dcomex  10404  axdc3lem4  10410  cnfldfunALT  21436  noextend  27727  nosupbday  27766  nosupbnd1  27775  nosupbnd2  27777  noinfbday  27781  noinfbnd1  27790  noinfbnd2  27792  bnj1416  35331  bnj1421  35334  fineqvac  35409  subfacp1lem2a  35527  subfacp1lem5  35531