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Theorem dmsnopss 6202
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 6201 . . 3 (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = {𝐴})
2 eqimss 4036 . . 3 (dom {⟨𝐴, 𝐵⟩} = {𝐴} → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴})
31, 2syl 17 . 2 (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴})
4 opprc2 4891 . . . . . 6 𝐵 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
54sneqd 4634 . . . . 5 𝐵 ∈ V → {⟨𝐴, 𝐵⟩} = {∅})
65dmeqd 5897 . . . 4 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = dom {∅})
7 dmsn0 6197 . . . 4 dom {∅} = ∅
86, 7eqtrdi 2787 . . 3 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = ∅)
9 0ss 4392 . . 3 ∅ ⊆ {𝐴}
108, 9eqsstrdi 4032 . 2 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴})
113, 10pm2.61i 182 1 dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1541  wcel 2106  Vcvv 3473  wss 3944  c0 4318  {csn 4622  cop 4628  dom cdm 5669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-xp 5675  df-dm 5679
This theorem is referenced by:  snopsuppss  8146  strle1  17073  setsres  17093  setscom  17095  setsid  17123  ex-res  29559  bj-fununsn1  35938  mapfzcons1  41226
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