Theorem dmsnopss 6166

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 6165	. . 3 ⊢ (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = {𝐴})
2		eqimss 3989	. . 3 ⊢ (dom {⟨𝐴, 𝐵⟩} = {𝐴} → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴})
3	1, 2	syl 17	. 2 ⊢ (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴})
4		opprc2 4849	. . . . . 6 ⊢ (¬ 𝐵 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
5	4	sneqd 4587	. . . . 5 ⊢ (¬ 𝐵 ∈ V → {⟨𝐴, 𝐵⟩} = {∅})
6	5	dmeqd 5849	. . . 4 ⊢ (¬ 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = dom {∅})
7		dmsn0 6161	. . . 4 ⊢ dom {∅} = ∅
8	6, 7	eqtrdi 2784	. . 3 ⊢ (¬ 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = ∅)
9		0ss 4349	. . 3 ⊢ ∅ ⊆ {𝐴}
10	8, 9	eqsstrdi 3975	. 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 2113  Vcvv 3437   ⊆ wss 3898  ∅c0 4282  {csn 4575  ⟨cop 4581  dom cdm 5619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-xp 5625  df-dm 5629

This theorem is referenced by:  snopsuppss  8115  strle1  17071  setsres  17091  setscom  17093  setsid  17120  ex-res  30423  bj-fununsn1  37318  mapfzcons1  42835