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Mirrors > Home > ILE Home > Th. List > dmsnopg | 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 2554 | . . . . . 6 ⊢ x ∈ V | |
2 | vex 2554 | . . . . . 6 ⊢ y ∈ V | |
3 | 1, 2 | opth1 3964 | . . . . 5 ⊢ (〈x, y〉 = 〈A, B〉 → x = A) |
4 | 3 | exlimiv 1486 | . . . 4 ⊢ (∃y〈x, y〉 = 〈A, B〉 → x = A) |
5 | opeq1 3540 | . . . . 5 ⊢ (x = A → 〈x, B〉 = 〈A, B〉) | |
6 | opeq2 3541 | . . . . . . 7 ⊢ (y = B → 〈x, y〉 = 〈x, B〉) | |
7 | 6 | eqeq1d 2045 | . . . . . 6 ⊢ (y = B → (〈x, y〉 = 〈A, B〉 ↔ 〈x, B〉 = 〈A, B〉)) |
8 | 7 | spcegv 2635 | . . . . 5 ⊢ (B ∈ 𝑉 → (〈x, B〉 = 〈A, B〉 → ∃y〈x, y〉 = 〈A, B〉)) |
9 | 5, 8 | syl5 28 | . . . 4 ⊢ (B ∈ 𝑉 → (x = A → ∃y〈x, y〉 = 〈A, B〉)) |
10 | 4, 9 | impbid2 131 | . . 3 ⊢ (B ∈ 𝑉 → (∃y〈x, y〉 = 〈A, B〉 ↔ x = A)) |
11 | 1 | eldm2 4476 | . . . 4 ⊢ (x ∈ dom {〈A, B〉} ↔ ∃y〈x, y〉 ∈ {〈A, B〉}) |
12 | 1, 2 | opex 3957 | . . . . . 6 ⊢ 〈x, y〉 ∈ V |
13 | 12 | elsnc 3390 | . . . . 5 ⊢ (〈x, y〉 ∈ {〈A, B〉} ↔ 〈x, y〉 = 〈A, B〉) |
14 | 13 | exbii 1493 | . . . 4 ⊢ (∃y〈x, y〉 ∈ {〈A, B〉} ↔ ∃y〈x, y〉 = 〈A, B〉) |
15 | 11, 14 | bitri 173 | . . 3 ⊢ (x ∈ dom {〈A, B〉} ↔ ∃y〈x, y〉 = 〈A, B〉) |
16 | elsn 3382 | . . 3 ⊢ (x ∈ {A} ↔ x = A) | |
17 | 10, 15, 16 | 3bitr4g 212 | . 2 ⊢ (B ∈ 𝑉 → (x ∈ dom {〈A, B〉} ↔ x ∈ {A})) |
18 | 17 | eqrdv 2035 | 1 ⊢ (B ∈ 𝑉 → dom {〈A, B〉} = {A}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 ∃wex 1378 ∈ wcel 1390 {csn 3367 〈cop 3370 dom cdm 4288 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-dm 4298 |
This theorem is referenced by: dmpropg 4736 dmsnop 4737 rnsnopg 4742 elxp4 4751 fnsng 4890 funprg 4892 funtpg 4893 fntpg 4898 |
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