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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 3476 | . . . . . 6 ⊢ 𝑥 ∈ V | |
2 | vex 3476 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | opth1 5474 | . . . . 5 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → 𝑥 = 𝐴) |
4 | 3 | exlimiv 1931 | . . . 4 ⊢ (∃𝑦⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → 𝑥 = 𝐴) |
5 | opeq1 4872 | . . . . 5 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝐵⟩ = ⟨𝐴, 𝐵⟩) | |
6 | opeq2 4873 | . . . . . . 7 ⊢ (𝑦 = 𝐵 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝐵⟩) | |
7 | 6 | eqeq1d 2732 | . . . . . 6 ⊢ (𝑦 = 𝐵 → (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ ⟨𝑥, 𝐵⟩ = ⟨𝐴, 𝐵⟩)) |
8 | 7 | spcegv 3586 | . . . . 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 5900 | . . . 4 ⊢ (𝑥 ∈ dom {⟨𝐴, 𝐵⟩} ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩}) |
12 | opex 5463 | . . . . . 6 ⊢ ⟨𝑥, 𝑦⟩ ∈ V | |
13 | 12 | elsn 4642 | . . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩) |
14 | 13 | exbii 1848 | . . . 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 4643 | . . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
17 | 10, 15, 16 | 3bitr4g 313 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ dom {⟨𝐴, 𝐵⟩} ↔ 𝑥 ∈ {𝐴})) |
18 | 17 | eqrdv 2728 | 1 ⊢ (𝐵 ∈ 𝑉 → dom {⟨𝐴, 𝐵⟩} = {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∃wex 1779 ∈ wcel 2104 {csn 4627 ⟨cop 4633 dom cdm 5675 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-dm 5685 |
This theorem is referenced by: dmsnopss 6212 dmpropg 6213 dmsnop 6214 rnsnopg 6219 fnsng 6599 funprg 6601 funtpg 6602 fntpg 6607 funsnfsupp 9389 s1dmALT 14563 setsval 17104 setsdm 17107 estrreslem2 18094 snstriedgval 28565 1loopgrvd0 29028 1hevtxdg0 29029 1hevtxdg1 29030 1egrvtxdg1 29033 p1evtxdeqlem 29036 wlkp1 29205 eupthp1 29736 trlsegvdeglem5 29744 cosnopne 32183 bnj96 34174 bnj535 34199 ovnovollem1 45670 |
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