| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > f1o2sn | Structured version Visualization version GIF version | ||
| Description: A singleton consisting in a nested ordered pair is a one-to-one function from the cartesian product of two singletons onto a singleton (case where the two singletons are equal). (Contributed by AV, 15-Aug-2019.) |
| Ref | Expression |
|---|---|
| f1o2sn | ⊢ ((𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → {〈〈𝐸, 𝐸〉, 𝑋〉}:({𝐸} × {𝐸})–1-1-onto→{𝑋}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | opex 5426 | . . 3 ⊢ 〈𝐸, 𝐸〉 ∈ V | |
| 2 | simpr 484 | . . 3 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → 𝑋 ∈ 𝑊) | |
| 3 | f1osng 6843 | . . 3 ⊢ ((〈𝐸, 𝐸〉 ∈ V ∧ 𝑋 ∈ 𝑊) → {〈〈𝐸, 𝐸〉, 𝑋〉}:{〈𝐸, 𝐸〉}–1-1-onto→{𝑋}) | |
| 4 | 1, 2, 3 | sylancr 587 | . 2 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → {〈〈𝐸, 𝐸〉, 𝑋〉}:{〈𝐸, 𝐸〉}–1-1-onto→{𝑋}) |
| 5 | xpsng 7113 | . . . . . 6 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝐸 ∈ 𝑉) → ({𝐸} × {𝐸}) = {〈𝐸, 𝐸〉}) | |
| 6 | 5 | anidms 566 | . . . . 5 ⊢ (𝐸 ∈ 𝑉 → ({𝐸} × {𝐸}) = {〈𝐸, 𝐸〉}) |
| 7 | 6 | eqcomd 2736 | . . . 4 ⊢ (𝐸 ∈ 𝑉 → {〈𝐸, 𝐸〉} = ({𝐸} × {𝐸})) |
| 8 | 7 | adantr 480 | . . 3 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → {〈𝐸, 𝐸〉} = ({𝐸} × {𝐸})) |
| 9 | 8 | f1oeq2d 6798 | . 2 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → ({〈〈𝐸, 𝐸〉, 𝑋〉}:{〈𝐸, 𝐸〉}–1-1-onto→{𝑋} ↔ {〈〈𝐸, 𝐸〉, 𝑋〉}:({𝐸} × {𝐸})–1-1-onto→{𝑋})) |
| 10 | 4, 9 | mpbid 232 | 1 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → {〈〈𝐸, 𝐸〉, 𝑋〉}:({𝐸} × {𝐸})–1-1-onto→{𝑋}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 Vcvv 3450 {csn 4591 〈cop 4597 × cxp 5638 –1-1-onto→wf1o 6512 |
| 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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5253 ax-nul 5263 ax-pr 5389 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-dif 3919 df-un 3921 df-ss 3933 df-nul 4299 df-if 4491 df-sn 4592 df-pr 4594 df-op 4598 df-br 5110 df-opab 5172 df-mpt 5191 df-id 5535 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 |
| This theorem is referenced by: mat1dimelbas 22364 |
| Copyright terms: Public domain | W3C validator |