Theorem dmsnopss 6197

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 6196	. . 3 ⊢ (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = {𝐴})
2		eqimss 3994	. . 3 ⊢ (dom {⟨𝐴, 𝐵⟩} = {𝐴} → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴})
3	1, 2	syl 17	. 2 ⊢ (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴})
4		opprc2 4855	. . . . . 6 ⊢ (¬ 𝐵 ∈ V → ⟨𝐴, 𝐵⟩ = ∅)
5	4	sneqd 4593	. . . . 5 ⊢ (¬ 𝐵 ∈ V → {⟨𝐴, 𝐵⟩} = {∅})
6	5	dmeqd 5879	. . . 4 ⊢ (¬ 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = dom {∅})
7		dmsn0 6192	. . . 4 ⊢ dom {∅} = ∅
8	6, 7	eqtrdi 2812	. . 3 ⊢ (¬ 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = ∅)
9		0ss 4353	. . 3 ⊢ ∅ ⊆ {𝐴}
10	8, 9	eqsstrdi 3980	. 2 ⊢ (¬ 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴})
11	3, 10	pm2.61i 183	1 ⊢ dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴}

Syntax hints:  ¬ wn 3   = wceq 1559   ∈ wcel 2141  Vcvv 3453   ⊆ wss 3904  ∅c0 4285  {csn 4581  ⟨cop 4587  dom cdm 5645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5651  df-dm 5655

This theorem is referenced by:  snopsuppss  8154  strle1  17177  setsres  17197  setscom  17199  setsid  17226  ex-res  30589  bj-fununsn1  37709  mapfzcons1  43262