Theorem dmsnop 6238

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 6235	. 2 ⊢ (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = {𝐴})
3	1, 2	ax-mp 5	1 ⊢ dom {⟨𝐴, 𝐵⟩} = {𝐴}

Syntax hints:   = wceq 1537   ∈ wcel 2106  Vcvv 3478  {csn 4631  ⟨cop 4637  dom cdm 5689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-dm 5699

This theorem is referenced by:  dmtpop  6240  dmsnsnsn  6242  op1sta  6247  snres0  6320  funtp  6625  funopdmsn  7170  frrlem14  8323  wfrlem13OLD  8360  wfrlem16OLD  8363  tfrlem10  8426  ac6sfi  9318  dcomex  10485  axdc3lem4  10491  cnfldfunALT  21397  cnfldfunALTOLD  21410  cnfldfunALTOLDOLD  21411  noextend  27726  nosupbday  27765  nosupbnd1  27774  nosupbnd2  27776  noinfbday  27780  noinfbnd1  27789  noinfbnd2  27791  bnj1416  35032  bnj1421  35035  fineqvac  35090  subfacp1lem2a  35165  subfacp1lem5  35169