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Mirrors > Home > MPE Home > Th. List > dmsnopg | Structured version Visualization version GIF version |
Description: The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
dmsnopg | ⊢ (𝐵 ∈ 𝑉 → dom {⟨𝐴, 𝐵⟩} = {𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3478 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | vex 3478 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | opth1 5475 | . . . . 5 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → 𝑥 = 𝐴) |
4 | 3 | exlimiv 1933 | . . . 4 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → 𝑥 = 𝐴) |
5 | opeq1 4873 | . . . . 5 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝐵⟩ = ⟨𝐴, 𝐵⟩) | |
6 | opeq2 4874 | . . . . . . 7 ⊢ (𝑦 = 𝐵 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝐵⟩) | |
7 | 6 | eqeq1d 2734 | . . . . . 6 ⊢ (𝑦 = 𝐵 → (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ ⟨𝑥, 𝐵⟩ = ⟨𝐴, 𝐵⟩)) |
8 | 7 | spcegv 3587 | . . . . 5 ⊢ (𝐵 ∈ 𝑉 → (⟨𝑥, 𝐵⟩ = ⟨𝐴, 𝐵⟩ → ∃𝑦⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)) |
9 | 5, 8 | syl5 34 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝑥 = 𝐴 → ∃𝑦⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)) |
10 | 4, 9 | impbid2 225 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (∃𝑦⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ 𝑥 = 𝐴)) |
11 | 1 | eldm2 5901 | . . . 4 ⊢ (𝑥 ∈ dom {⟨𝐴, 𝐵⟩} ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩}) |
12 | opex 5464 | . . . . . 6 ⊢ ⟨𝑥, 𝑦⟩ ∈ V | |
13 | 12 | elsn 4643 | . . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩) |
14 | 13 | exbii 1850 | . . . 4 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ∃𝑦⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩) |
15 | 11, 14 | bitri 274 | . . 3 ⊢ (𝑥 ∈ dom {⟨𝐴, 𝐵⟩} ↔ ∃𝑦⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩) |
16 | velsn 4644 | . . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
17 | 10, 15, 16 | 3bitr4g 313 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ dom {⟨𝐴, 𝐵⟩} ↔ 𝑥 ∈ {𝐴})) |
18 | 17 | eqrdv 2730 | 1 ⊢ (𝐵 ∈ 𝑉 → dom {⟨𝐴, 𝐵⟩} = {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∃wex 1781 ∈ wcel 2106 {csn 4628 ⟨cop 4634 dom cdm 5676 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-dm 5686 |
This theorem is referenced by: dmsnopss 6213 dmpropg 6214 dmsnop 6215 rnsnopg 6220 fnsng 6600 funprg 6602 funtpg 6603 fntpg 6608 funsnfsupp 9386 s1dmALT 14558 setsval 17099 setsdm 17102 estrreslem2 18089 snstriedgval 28295 1loopgrvd0 28758 1hevtxdg0 28759 1hevtxdg1 28760 1egrvtxdg1 28763 p1evtxdeqlem 28766 wlkp1 28935 eupthp1 29466 trlsegvdeglem5 29474 cosnopne 31911 bnj96 33871 bnj535 33896 ovnovollem1 45362 |
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