Theorem dmsnopss 6106

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 6105	. . 3 ⊢ (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = {𝐴})
2		eqimss 3973	. . 3 ⊢ (dom {⟨𝐴, 𝐵⟩} = {𝐴} → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴})
3	1, 2	syl 17	. 2 ⊢ (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴})
4		opprc2 4826	. . . . . 6 ⊢ (¬ 𝐵 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
5	4	sneqd 4570	. . . . 5 ⊢ (¬ 𝐵 ∈ V → {⟨𝐴, 𝐵⟩} = {∅})
6	5	dmeqd 5803	. . . 4 ⊢ (¬ 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = dom {∅})
7		dmsn0 6101	. . . 4 ⊢ dom {∅} = ∅
8	6, 7	eqtrdi 2795	. . 3 ⊢ (¬ 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = ∅)
9		0ss 4327	. . 3 ⊢ ∅ ⊆ {𝐴}
10	8, 9	eqsstrdi 3971	. 2 ⊢ (¬ 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴})
11	3, 10	pm2.61i 182	1 ⊢ dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴}

Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ⊆ wss 3883  ∅c0 4253  {csn 4558  ⟨cop 4564  dom cdm 5580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-dm 5590

This theorem is referenced by:  snopsuppss  7966  strle1  16787  setsres  16807  setscom  16809  setsid  16837  ex-res  28706  bj-fununsn1  35351  mapfzcons1  40455