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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 3484 | . . . . . 6 ⊢ 𝑥 ∈ V | |
| 2 | vex 3484 | . . . . . 6 ⊢ 𝑦 ∈ V | |
| 3 | 1, 2 | opth1 5480 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉 → 𝑥 = 𝐴) |
| 4 | 3 | exlimiv 1930 | . . . 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 2739 | . . . . . 6 ⊢ (𝑦 = 𝐵 → (〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉 ↔ 〈𝑥, 𝐵〉 = 〈𝐴, 𝐵〉)) |
| 8 | 7 | spcegv 3597 | . . . . 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 226 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (∃𝑦〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉 ↔ 𝑥 = 𝐴)) |
| 11 | 1 | eldm2 5912 | . . . 4 ⊢ (𝑥 ∈ dom {〈𝐴, 𝐵〉} ↔ ∃𝑦〈𝑥, 𝑦〉 ∈ {〈𝐴, 𝐵〉}) |
| 12 | opex 5469 | . . . . . 6 ⊢ 〈𝑥, 𝑦〉 ∈ V | |
| 13 | 12 | elsn 4641 | . . . . 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 275 | . . 3 ⊢ (𝑥 ∈ dom {〈𝐴, 𝐵〉} ↔ ∃𝑦〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉) |
| 16 | velsn 4642 | . . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
| 17 | 10, 15, 16 | 3bitr4g 314 | . 2 ⊢ (𝐵 ∈ 𝑉 → (𝑥 ∈ dom {〈𝐴, 𝐵〉} ↔ 𝑥 ∈ {𝐴})) |
| 18 | 17 | eqrdv 2735 | 1 ⊢ (𝐵 ∈ 𝑉 → dom {〈𝐴, 𝐵〉} = {𝐴}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∃wex 1779 ∈ wcel 2108 {csn 4626 〈cop 4632 dom cdm 5685 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-dm 5695 |
| This theorem is referenced by: dmsnopss 6234 dmpropg 6235 dmsnop 6236 rnsnopg 6241 fnsng 6618 funprg 6620 funtpg 6621 fntpg 6626 funsnfsupp 9432 s1dmALT 14647 setsval 17204 setsdm 17207 estrreslem2 18183 snstriedgval 29055 1loopgrvd0 29522 1hevtxdg0 29523 1hevtxdg1 29524 1egrvtxdg1 29527 p1evtxdeqlem 29530 wlkp1 29699 eupthp1 30235 trlsegvdeglem5 30243 cosnopne 32703 bnj96 34879 bnj535 34904 ovnovollem1 46671 |
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