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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 5347 | . . 3 ⊢ 〈𝐸, 𝐸〉 ∈ V | |
2 | simpr 485 | . . 3 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → 𝑋 ∈ 𝑊) | |
3 | f1osng 6648 | . . 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 6893 | . . . . . 6 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝐸 ∈ 𝑉) → ({𝐸} × {𝐸}) = {〈𝐸, 𝐸〉}) | |
6 | 5 | anidms 567 | . . . . 5 ⊢ (𝐸 ∈ 𝑉 → ({𝐸} × {𝐸}) = {〈𝐸, 𝐸〉}) |
7 | 6 | eqcomd 2824 | . . . 4 ⊢ (𝐸 ∈ 𝑉 → {〈𝐸, 𝐸〉} = ({𝐸} × {𝐸})) |
8 | 7 | adantr 481 | . . 3 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → {〈𝐸, 𝐸〉} = ({𝐸} × {𝐸})) |
9 | 8 | f1oeq2d 6604 | . 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 233 | 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 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 {csn 4557 〈cop 4563 × cxp 5546 –1-1-onto→wf1o 6347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 |
This theorem is referenced by: mat1dimelbas 21008 |
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