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Mirrors > Home > MPE Home > Th. List > f1o2sn | Structured version Visualization version GIF version |
Description: A singleton with a nested ordered pair is a 1-1 function of the cartesian product of two singleton onto a singleton. (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 4859 | . . 3 ⊢ 〈𝐸, 𝐸〉 ∈ V | |
2 | simpr 476 | . . 3 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → 𝑋 ∈ 𝑊) | |
3 | f1osng 6089 | . . 3 ⊢ ((〈𝐸, 𝐸〉 ∈ V ∧ 𝑋 ∈ 𝑊) → {〈〈𝐸, 𝐸〉, 𝑋〉}:{〈𝐸, 𝐸〉}–1-1-onto→{𝑋}) | |
4 | 1, 2, 3 | sylancr 694 | . 2 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → {〈〈𝐸, 𝐸〉, 𝑋〉}:{〈𝐸, 𝐸〉}–1-1-onto→{𝑋}) |
5 | xpsng 6312 | . . . . . 6 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝐸 ∈ 𝑉) → ({𝐸} × {𝐸}) = {〈𝐸, 𝐸〉}) | |
6 | 5 | anidms 675 | . . . . 5 ⊢ (𝐸 ∈ 𝑉 → ({𝐸} × {𝐸}) = {〈𝐸, 𝐸〉}) |
7 | 6 | eqcomd 2616 | . . . 4 ⊢ (𝐸 ∈ 𝑉 → {〈𝐸, 𝐸〉} = ({𝐸} × {𝐸})) |
8 | 7 | adantr 480 | . . 3 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → {〈𝐸, 𝐸〉} = ({𝐸} × {𝐸})) |
9 | f1oeq2 6041 | . . 3 ⊢ ({〈𝐸, 𝐸〉} = ({𝐸} × {𝐸}) → ({〈〈𝐸, 𝐸〉, 𝑋〉}:{〈𝐸, 𝐸〉}–1-1-onto→{𝑋} ↔ {〈〈𝐸, 𝐸〉, 𝑋〉}:({𝐸} × {𝐸})–1-1-onto→{𝑋})) | |
10 | 8, 9 | syl 17 | . 2 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → ({〈〈𝐸, 𝐸〉, 𝑋〉}:{〈𝐸, 𝐸〉}–1-1-onto→{𝑋} ↔ {〈〈𝐸, 𝐸〉, 𝑋〉}:({𝐸} × {𝐸})–1-1-onto→{𝑋})) |
11 | 4, 10 | mpbid 221 | 1 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → {〈〈𝐸, 𝐸〉, 𝑋〉}:({𝐸} × {𝐸})–1-1-onto→{𝑋}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {csn 4125 〈cop 4131 × cxp 5036 –1-1-onto→wf1o 5803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 |
This theorem is referenced by: mat1dimelbas 20096 |
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