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Theorem dmsnop 6172
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
StepHypRef Expression
1 dmsnop.1 . 2 𝐵 ∈ V
2 dmsnopg 6169 . 2 (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = {𝐴})
31, 2ax-mp 5 1 dom {⟨𝐴, 𝐵⟩} = {𝐴}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  Vcvv 3447  {csn 4590  cop 4596  dom cdm 5637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-dm 5647
This theorem is referenced by:  dmtpop  6174  dmsnsnsn  6176  op1sta  6181  snres0  6254  funtp  6562  funopdmsn  7100  frrlem14  8234  wfrlem13OLD  8271  wfrlem16OLD  8274  tfrlem10  8337  ac6sfi  9237  dcomex  10391  axdc3lem4  10397  cnfldfunALT  20832  cnfldfunALTOLD  20833  noextend  27037  nosupbday  27076  nosupbnd1  27085  nosupbnd2  27087  noinfbday  27091  noinfbnd1  27100  noinfbnd2  27102  bnj1416  33715  bnj1421  33718  fineqvac  33762  subfacp1lem2a  33838  subfacp1lem5  33842
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