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| 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 5469 | . . 3 ⊢ 〈𝐸, 𝐸〉 ∈ V | |
| 2 | simpr 484 | . . 3 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → 𝑋 ∈ 𝑊) | |
| 3 | f1osng 6889 | . . 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 7159 | . . . . . 6 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝐸 ∈ 𝑉) → ({𝐸} × {𝐸}) = {〈𝐸, 𝐸〉}) | |
| 6 | 5 | anidms 566 | . . . . 5 ⊢ (𝐸 ∈ 𝑉 → ({𝐸} × {𝐸}) = {〈𝐸, 𝐸〉}) | 
| 7 | 6 | eqcomd 2743 | . . . 4 ⊢ (𝐸 ∈ 𝑉 → {〈𝐸, 𝐸〉} = ({𝐸} × {𝐸})) | 
| 8 | 7 | adantr 480 | . . 3 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → {〈𝐸, 𝐸〉} = ({𝐸} × {𝐸})) | 
| 9 | 8 | f1oeq2d 6844 | . 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 2108 Vcvv 3480 {csn 4626 〈cop 4632 × cxp 5683 –1-1-onto→wf1o 6560 | 
| 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-10 2141 ax-11 2157 ax-12 2177 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-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 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-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 | 
| This theorem is referenced by: mat1dimelbas 22477 | 
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