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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 45011 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ {𝐴} ↦ 𝐵) = ({𝐴} × {𝐵})) | |
2 | snex 5454 | . . . 4 ⊢ {𝐴} ∈ V | |
3 | distopon 23018 | . . . 4 ⊢ ({𝐴} ∈ V → 𝒫 {𝐴} ∈ (TopOn‘{𝐴})) | |
4 | 2, 3 | mp1i 13 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 {𝐴} ∈ (TopOn‘{𝐴})) |
5 | snex 5454 | . . . 4 ⊢ {𝐵} ∈ V | |
6 | distopon 23018 | . . . 4 ⊢ ({𝐵} ∈ V → 𝒫 {𝐵} ∈ (TopOn‘{𝐵})) | |
7 | 5, 6 | mp1i 13 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 {𝐵} ∈ (TopOn‘{𝐵})) |
8 | snidg 4682 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐵}) | |
9 | 8 | adantl 481 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ {𝐵}) |
10 | cnconst2 23305 | . . 3 ⊢ ((𝒫 {𝐴} ∈ (TopOn‘{𝐴}) ∧ 𝒫 {𝐵} ∈ (TopOn‘{𝐵}) ∧ 𝐵 ∈ {𝐵}) → ({𝐴} × {𝐵}) ∈ (𝒫 {𝐴} Cn 𝒫 {𝐵})) | |
11 | 4, 7, 9, 10 | syl3anc 1371 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} × {𝐵}) ∈ (𝒫 {𝐴} Cn 𝒫 {𝐵})) |
12 | 1, 11 | eqeltrd 2838 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ {𝐴} ↦ 𝐵) ∈ (𝒫 {𝐴} Cn 𝒫 {𝐵})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2103 Vcvv 3482 𝒫 cpw 4622 {csn 4648 ↦ cmpt 5252 × cxp 5697 ‘cfv 6572 (class class class)co 7445 TopOnctopon 22930 Cn ccn 23246 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-ov 7448 df-oprab 7449 df-mpo 7450 df-1st 8026 df-2nd 8027 df-map 8882 df-topgen 17498 df-top 22914 df-topon 22931 df-cn 23249 df-cnp 23250 |
This theorem is referenced by: cncfdmsn 45746 |
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