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Theorem dmsnopss 6064
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 6063 . . 3 (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = {𝐴})
2 eqimss 4020 . . 3 (dom {⟨𝐴, 𝐵⟩} = {𝐴} → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴})
31, 2syl 17 . 2 (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴})
4 opprc2 4820 . . . . . 6 𝐵 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
54sneqd 4569 . . . . 5 𝐵 ∈ V → {⟨𝐴, 𝐵⟩} = {∅})
65dmeqd 5767 . . . 4 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = dom {∅})
7 dmsn0 6059 . . . 4 dom {∅} = ∅
86, 7syl6eq 2869 . . 3 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = ∅)
9 0ss 4347 . . 3 ∅ ⊆ {𝐴}
108, 9eqsstrdi 4018 . 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 1528  wcel 2105  Vcvv 3492  wss 3933  c0 4288  {csn 4557  cop 4563  dom cdm 5548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-dm 5558
This theorem is referenced by:  snopsuppss  7834  setsres  16513  setscom  16515  setsid  16526  strle1  16580  ex-res  28147  bj-fununsn1  34427  mapfzcons1  39192
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