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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnfdmsn | Structured version Visualization version GIF version |
Description: A function with a singleton domain is continuous. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
cnfdmsn | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ {𝐴} ↦ 𝐵) ∈ (𝒫 {𝐴} Cn 𝒫 {𝐵})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fmptsnxp 40093 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ {𝐴} ↦ 𝐵) = ({𝐴} × {𝐵})) | |
2 | snex 5098 | . . . 4 ⊢ {𝐴} ∈ V | |
3 | distopon 21127 | . . . 4 ⊢ ({𝐴} ∈ V → 𝒫 {𝐴} ∈ (TopOn‘{𝐴})) | |
4 | 2, 3 | mp1i 13 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 {𝐴} ∈ (TopOn‘{𝐴})) |
5 | snex 5098 | . . . 4 ⊢ {𝐵} ∈ V | |
6 | distopon 21127 | . . . 4 ⊢ ({𝐵} ∈ V → 𝒫 {𝐵} ∈ (TopOn‘{𝐵})) | |
7 | 5, 6 | mp1i 13 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 {𝐵} ∈ (TopOn‘{𝐵})) |
8 | snidg 4397 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐵}) | |
9 | 8 | adantl 474 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ {𝐵}) |
10 | cnconst2 21413 | . . 3 ⊢ ((𝒫 {𝐴} ∈ (TopOn‘{𝐴}) ∧ 𝒫 {𝐵} ∈ (TopOn‘{𝐵}) ∧ 𝐵 ∈ {𝐵}) → ({𝐴} × {𝐵}) ∈ (𝒫 {𝐴} Cn 𝒫 {𝐵})) | |
11 | 4, 7, 9, 10 | syl3anc 1491 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} × {𝐵}) ∈ (𝒫 {𝐴} Cn 𝒫 {𝐵})) |
12 | 1, 11 | eqeltrd 2877 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ {𝐴} ↦ 𝐵) ∈ (𝒫 {𝐴} Cn 𝒫 {𝐵})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∈ wcel 2157 Vcvv 3384 𝒫 cpw 4348 {csn 4367 ↦ cmpt 4921 × cxp 5309 ‘cfv 6100 (class class class)co 6877 TopOnctopon 21040 Cn ccn 21354 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2776 ax-sep 4974 ax-nul 4982 ax-pow 5034 ax-pr 5096 ax-un 7182 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2785 df-cleq 2791 df-clel 2794 df-nfc 2929 df-ne 2971 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3386 df-sbc 3633 df-csb 3728 df-dif 3771 df-un 3773 df-in 3775 df-ss 3782 df-nul 4115 df-if 4277 df-pw 4350 df-sn 4368 df-pr 4370 df-op 4374 df-uni 4628 df-iun 4711 df-br 4843 df-opab 4905 df-mpt 4922 df-id 5219 df-xp 5317 df-rel 5318 df-cnv 5319 df-co 5320 df-dm 5321 df-rn 5322 df-res 5323 df-ima 5324 df-iota 6063 df-fun 6102 df-fn 6103 df-f 6104 df-f1 6105 df-fo 6106 df-f1o 6107 df-fv 6108 df-ov 6880 df-oprab 6881 df-mpt2 6882 df-1st 7400 df-2nd 7401 df-map 8096 df-topgen 16416 df-top 21024 df-topon 21041 df-cn 21357 df-cnp 21358 |
This theorem is referenced by: cncfdmsn 40836 |
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