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Mirrors > Home > ILE Home > Th. List > dmsnsnsng | GIF version |
Description: The domain of the singleton of the singleton of a singleton. (Contributed by Jim Kingdon, 16-Dec-2018.) |
Ref | Expression |
---|---|
dmsnsnsng | ⊢ (𝐴 ∈ V → dom {{{𝐴}}} = {𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2684 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
2 | 1 | opid 3718 | . . . . . 6 ⊢ 〈𝑥, 𝑥〉 = {{𝑥}} |
3 | sneq 3533 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
4 | 3 | sneqd 3535 | . . . . . 6 ⊢ (𝑥 = 𝐴 → {{𝑥}} = {{𝐴}}) |
5 | 2, 4 | syl5eq 2182 | . . . . 5 ⊢ (𝑥 = 𝐴 → 〈𝑥, 𝑥〉 = {{𝐴}}) |
6 | 5 | sneqd 3535 | . . . 4 ⊢ (𝑥 = 𝐴 → {〈𝑥, 𝑥〉} = {{{𝐴}}}) |
7 | 6 | dmeqd 4736 | . . 3 ⊢ (𝑥 = 𝐴 → dom {〈𝑥, 𝑥〉} = dom {{{𝐴}}}) |
8 | 7, 3 | eqeq12d 2152 | . 2 ⊢ (𝑥 = 𝐴 → (dom {〈𝑥, 𝑥〉} = {𝑥} ↔ dom {{{𝐴}}} = {𝐴})) |
9 | 1 | dmsnop 5007 | . 2 ⊢ dom {〈𝑥, 𝑥〉} = {𝑥} |
10 | 8, 9 | vtoclg 2741 | 1 ⊢ (𝐴 ∈ V → dom {{{𝐴}}} = {𝐴}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 Vcvv 2681 {csn 3522 〈cop 3525 dom cdm 4534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-dm 4544 |
This theorem is referenced by: (None) |
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