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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 5356 | . . 3 ⊢ 〈𝐸, 𝐸〉 ∈ V | |
2 | simpr 487 | . . 3 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → 𝑋 ∈ 𝑊) | |
3 | f1osng 6655 | . . 3 ⊢ ((〈𝐸, 𝐸〉 ∈ V ∧ 𝑋 ∈ 𝑊) → {〈〈𝐸, 𝐸〉, 𝑋〉}:{〈𝐸, 𝐸〉}–1-1-onto→{𝑋}) | |
4 | 1, 2, 3 | sylancr 589 | . 2 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → {〈〈𝐸, 𝐸〉, 𝑋〉}:{〈𝐸, 𝐸〉}–1-1-onto→{𝑋}) |
5 | xpsng 6901 | . . . . . 6 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝐸 ∈ 𝑉) → ({𝐸} × {𝐸}) = {〈𝐸, 𝐸〉}) | |
6 | 5 | anidms 569 | . . . . 5 ⊢ (𝐸 ∈ 𝑉 → ({𝐸} × {𝐸}) = {〈𝐸, 𝐸〉}) |
7 | 6 | eqcomd 2827 | . . . 4 ⊢ (𝐸 ∈ 𝑉 → {〈𝐸, 𝐸〉} = ({𝐸} × {𝐸})) |
8 | 7 | adantr 483 | . . 3 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → {〈𝐸, 𝐸〉} = ({𝐸} × {𝐸})) |
9 | 8 | f1oeq2d 6611 | . 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 234 | 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 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 {csn 4567 〈cop 4573 × cxp 5553 –1-1-onto→wf1o 6354 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 |
This theorem is referenced by: mat1dimelbas 21080 |
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