Theorem dmsnopss 6172

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 6171	. . 3 ⊢ (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = {𝐴})
2		eqimss 3981	. . 3 ⊢ (dom {⟨𝐴, 𝐵⟩} = {𝐴} → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴})
3	1, 2	syl 17	. 2 ⊢ (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴})
4		opprc2 4842	. . . . . 6 ⊢ (¬ 𝐵 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
5	4	sneqd 4580	. . . . 5 ⊢ (¬ 𝐵 ∈ V → {⟨𝐴, 𝐵⟩} = {∅})
6	5	dmeqd 5854	. . . 4 ⊢ (¬ 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = dom {∅})
7		dmsn0 6167	. . . 4 ⊢ dom {∅} = ∅
8	6, 7	eqtrdi 2788	. . 3 ⊢ (¬ 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = ∅)
9		0ss 4341	. . 3 ⊢ ∅ ⊆ {𝐴}
10	8, 9	eqsstrdi 3967	. 2 ⊢ (¬ 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴})
11	3, 10	pm2.61i 182	1 ⊢ dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴}

Syntax hints:  ¬ wn 3   = wceq 1542   ∈ wcel 2114  Vcvv 3430   ⊆ wss 3890  ∅c0 4274  {csn 4568  ⟨cop 4574  dom cdm 5624

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5630  df-dm 5634

This theorem is referenced by:  snopsuppss  8122  strle1  17119  setsres  17139  setscom  17141  setsid  17168  ex-res  30526  bj-fununsn1  37583  mapfzcons1  43163