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Theorem dmsnopss 5605
 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 5604 . . 3 (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = {𝐴})
2 eqimss 3655 . . 3 (dom {⟨𝐴, 𝐵⟩} = {𝐴} → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴})
31, 2syl 17 . 2 (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴})
4 opprc2 4424 . . . . . 6 𝐵 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
54sneqd 4187 . . . . 5 𝐵 ∈ V → {⟨𝐴, 𝐵⟩} = {∅})
65dmeqd 5324 . . . 4 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = dom {∅})
7 dmsn0 5600 . . . 4 dom {∅} = ∅
86, 7syl6eq 2671 . . 3 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = ∅)
9 0ss 3970 . . 3 ∅ ⊆ {𝐴}
108, 9syl6eqss 3653 . 2 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴})
113, 10pm2.61i 176 1 dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1482   ∈ wcel 1989  Vcvv 3198   ⊆ wss 3572  ∅c0 3913  {csn 4175  ⟨cop 4181  dom cdm 5112 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-br 4652  df-opab 4711  df-xp 5118  df-dm 5122 This theorem is referenced by:  snopsuppss  7307  setsres  15895  setscom  15897  setsid  15908  strlemor1OLD  15963  strle1  15967  ex-res  27282  mapfzcons1  37106
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