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Mirrors > Home > MPE Home > Th. List > dmsnopss | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
dmsnopss | ⊢ dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmsnopg 6169 | . . 3 ⊢ (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = {𝐴}) | |
2 | eqimss 4004 | . . 3 ⊢ (dom {⟨𝐴, 𝐵⟩} = {𝐴} → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴}) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴}) |
4 | opprc2 4859 | . . . . . 6 ⊢ (¬ 𝐵 ∈ V → ⟨𝐴, 𝐵⟩ = ∅) | |
5 | 4 | sneqd 4602 | . . . . 5 ⊢ (¬ 𝐵 ∈ V → {⟨𝐴, 𝐵⟩} = {∅}) |
6 | 5 | dmeqd 5865 | . . . 4 ⊢ (¬ 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = dom {∅}) |
7 | dmsn0 6165 | . . . 4 ⊢ dom {∅} = ∅ | |
8 | 6, 7 | eqtrdi 2789 | . . 3 ⊢ (¬ 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} = ∅) |
9 | 0ss 4360 | . . 3 ⊢ ∅ ⊆ {𝐴} | |
10 | 8, 9 | eqsstrdi 4002 | . 2 ⊢ (¬ 𝐵 ∈ V → dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴}) |
11 | 3, 10 | pm2.61i 182 | 1 ⊢ dom {⟨𝐴, 𝐵⟩} ⊆ {𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2107 Vcvv 3447 ⊆ wss 3914 ∅c0 4286 {csn 4590 ⟨cop 4596 dom cdm 5637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-xp 5643 df-dm 5647 |
This theorem is referenced by: snopsuppss 8114 strle1 17038 setsres 17058 setscom 17060 setsid 17088 ex-res 29434 bj-fununsn1 35774 mapfzcons1 41087 |
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