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Theorem dmsnop 5768
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 5765 . 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 1632  wcel 2139  Vcvv 3340  {csn 4321  cop 4327  dom cdm 5266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-dm 5276
This theorem is referenced by:  dmtpop  5770  dmsnsnsn  5772  op1sta  5777  funtp  6106  funopdmsn  6579  wfrlem13  7597  wfrlem16  7600  tfrlem10  7653  ac6sfi  8371  dcomex  9481  axdc3lem4  9487  cnfldfun  19980  bnj1416  31435  bnj1421  31438  subfacp1lem2a  31490  subfacp1lem5  31494  noextend  32146  nosupbday  32178  nosupbnd1  32187  nosupbnd2  32189
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