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Theorem dmsnopss 6068
Description: The domain of a singleton of an ordered pair is a subset of the singleton of the first member (with no sethood assumptions on 𝐵). (Contributed by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
dmsnopss dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴}

Proof of Theorem dmsnopss
StepHypRef Expression
1 dmsnopg 6067 . . 3 (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = {𝐴})
2 eqimss 4026 . . 3 (dom {⟨𝐴, 𝐵⟩} = {𝐴} → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴})
31, 2syl 17 . 2 (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴})
4 opprc2 4826 . . . . . 6 𝐵 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
54sneqd 4575 . . . . 5 𝐵 ∈ V → {⟨𝐴, 𝐵⟩} = {∅})
65dmeqd 5772 . . . 4 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = dom {∅})
7 dmsn0 6063 . . . 4 dom {∅} = ∅
86, 7syl6eq 2876 . . 3 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = ∅)
9 0ss 4353 . . 3 ∅ ⊆ {𝐴}
108, 9eqsstrdi 4024 . 2 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴})
113, 10pm2.61i 183 1 dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1530  wcel 2106  Vcvv 3499  wss 3939  c0 4294  {csn 4563  cop 4569  dom cdm 5553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-br 5063  df-opab 5125  df-xp 5559  df-dm 5563
This theorem is referenced by:  snopsuppss  7839  setsres  16517  setscom  16519  setsid  16530  strle1  16584  ex-res  28135  bj-fununsn1  34415  mapfzcons1  39176
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