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

Step	Hyp	Ref	Expression
1		dmsnopg 6201	. . 3 ⊢ (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = {𝐴})
2		eqimss 4036	. . 3 ⊢ (dom {⟨𝐴, 𝐵⟩} = {𝐴} → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴})
3	1, 2	syl 17	. 2 ⊢ (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴})
4		opprc2 4891	. . . . . 6 ⊢ (¬ 𝐵 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
5	4	sneqd 4634	. . . . 5 ⊢ (¬ 𝐵 ∈ V → {⟨𝐴, 𝐵⟩} = {∅})
6	5	dmeqd 5897	. . . 4 ⊢ (¬ 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = dom {∅})
7		dmsn0 6197	. . . 4 ⊢ dom {∅} = ∅
8	6, 7	eqtrdi 2787	. . 3 ⊢ (¬ 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = ∅)
9		0ss 4392	. . 3 ⊢ ∅ ⊆ {𝐴}
10	8, 9	eqsstrdi 4032	. 2 ⊢ (¬ 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴})
11	3, 10	pm2.61i 182	1 ⊢ dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴}

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