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 41417 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ {𝐴} ↦ 𝐵) = ({𝐴} × {𝐵})) | |
2 | snex 5324 | . . . 4 ⊢ {𝐴} ∈ V | |
3 | distopon 21599 | . . . 4 ⊢ ({𝐴} ∈ V → 𝒫 {𝐴} ∈ (TopOn‘{𝐴})) | |
4 | 2, 3 | mp1i 13 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 {𝐴} ∈ (TopOn‘{𝐴})) |
5 | snex 5324 | . . . 4 ⊢ {𝐵} ∈ V | |
6 | distopon 21599 | . . . 4 ⊢ ({𝐵} ∈ V → 𝒫 {𝐵} ∈ (TopOn‘{𝐵})) | |
7 | 5, 6 | mp1i 13 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 {𝐵} ∈ (TopOn‘{𝐵})) |
8 | snidg 4593 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ {𝐵}) | |
9 | 8 | adantl 484 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ {𝐵}) |
10 | cnconst2 21885 | . . 3 ⊢ ((𝒫 {𝐴} ∈ (TopOn‘{𝐴}) ∧ 𝒫 {𝐵} ∈ (TopOn‘{𝐵}) ∧ 𝐵 ∈ {𝐵}) → ({𝐴} × {𝐵}) ∈ (𝒫 {𝐴} Cn 𝒫 {𝐵})) | |
11 | 4, 7, 9, 10 | syl3anc 1367 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({𝐴} × {𝐵}) ∈ (𝒫 {𝐴} Cn 𝒫 {𝐵})) |
12 | 1, 11 | eqeltrd 2913 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ {𝐴} ↦ 𝐵) ∈ (𝒫 {𝐴} Cn 𝒫 {𝐵})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 Vcvv 3495 𝒫 cpw 4539 {csn 4561 ↦ cmpt 5139 × cxp 5548 ‘cfv 6350 (class class class)co 7150 TopOnctopon 21512 Cn ccn 21826 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-map 8402 df-topgen 16711 df-top 21496 df-topon 21513 df-cn 21829 df-cnp 21830 |
This theorem is referenced by: cncfdmsn 42165 |
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