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Theorem dmsnopss 6048
 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 6047 . . 3 (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = {𝐴})
2 eqimss 3950 . . 3 (dom {⟨𝐴, 𝐵⟩} = {𝐴} → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴})
31, 2syl 17 . 2 (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴})
4 opprc2 4791 . . . . . 6 𝐵 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
54sneqd 4537 . . . . 5 𝐵 ∈ V → {⟨𝐴, 𝐵⟩} = {∅})
65dmeqd 5751 . . . 4 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = dom {∅})
7 dmsn0 6043 . . . 4 dom {∅} = ∅
86, 7eqtrdi 2809 . . 3 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = ∅)
9 0ss 4295 . . 3 ∅ ⊆ {𝐴}
108, 9eqsstrdi 3948 . 2 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴})
113, 10pm2.61i 185 1 dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1538   ∈ wcel 2111  Vcvv 3409   ⊆ wss 3860  ∅c0 4227  {csn 4525  ⟨cop 4531  dom cdm 5528 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ne 2952  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5037  df-opab 5099  df-xp 5534  df-dm 5538 This theorem is referenced by:  snopsuppss  7859  setsres  16596  setscom  16598  setsid  16609  strle1  16663  ex-res  28338  bj-fununsn1  34982  mapfzcons1  40066
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