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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 39869 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ {𝐴} ↦ 𝐵) = ({𝐴} × {𝐵})) | |
2 | snex 5036 | . . . 4 ⊢ {𝐴} ∈ V | |
3 | distopon 21022 | . . . 4 ⊢ ({𝐴} ∈ V → 𝒫 {𝐴} ∈ (TopOn‘{𝐴})) | |
4 | 2, 3 | mp1i 13 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 {𝐴} ∈ (TopOn‘{𝐴})) |
5 | snex 5036 | . . . 4 ⊢ {𝐵} ∈ V | |
6 | distopon 21022 | . . . 4 ⊢ ({𝐵} ∈ V → 𝒫 {𝐵} ∈ (TopOn‘{𝐵})) | |
7 | 5, 6 | mp1i 13 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 {𝐵} ∈ (TopOn‘{𝐵})) |
8 | snidg 4345 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐵}) | |
9 | 8 | adantl 467 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ {𝐵}) |
10 | cnconst2 21308 | . . 3 ⊢ ((𝒫 {𝐴} ∈ (TopOn‘{𝐴}) ∧ 𝒫 {𝐵} ∈ (TopOn‘{𝐵}) ∧ 𝐵 ∈ {𝐵}) → ({𝐴} × {𝐵}) ∈ (𝒫 {𝐴} Cn 𝒫 {𝐵})) | |
11 | 4, 7, 9, 10 | syl3anc 1476 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} × {𝐵}) ∈ (𝒫 {𝐴} Cn 𝒫 {𝐵})) |
12 | 1, 11 | eqeltrd 2850 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ {𝐴} ↦ 𝐵) ∈ (𝒫 {𝐴} Cn 𝒫 {𝐵})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∈ wcel 2145 Vcvv 3351 𝒫 cpw 4297 {csn 4316 ↦ cmpt 4863 × cxp 5247 ‘cfv 6031 (class class class)co 6793 TopOnctopon 20935 Cn ccn 21249 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-1st 7315 df-2nd 7316 df-map 8011 df-topgen 16312 df-top 20919 df-topon 20936 df-cn 21252 df-cnp 21253 |
This theorem is referenced by: cncfdmsn 40621 |
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