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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 | β’ ((π΄ β π β§ π΅ β π) β (π₯ β {π΄} β¦ π΅) β (π« {π΄} Cn π« {π΅})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fmptsnxp 43478 | . 2 β’ ((π΄ β π β§ π΅ β π) β (π₯ β {π΄} β¦ π΅) = ({π΄} Γ {π΅})) | |
2 | snex 5392 | . . . 4 β’ {π΄} β V | |
3 | distopon 22370 | . . . 4 β’ ({π΄} β V β π« {π΄} β (TopOnβ{π΄})) | |
4 | 2, 3 | mp1i 13 | . . 3 β’ ((π΄ β π β§ π΅ β π) β π« {π΄} β (TopOnβ{π΄})) |
5 | snex 5392 | . . . 4 β’ {π΅} β V | |
6 | distopon 22370 | . . . 4 β’ ({π΅} β V β π« {π΅} β (TopOnβ{π΅})) | |
7 | 5, 6 | mp1i 13 | . . 3 β’ ((π΄ β π β§ π΅ β π) β π« {π΅} β (TopOnβ{π΅})) |
8 | snidg 4624 | . . . 4 β’ (π΅ β π β π΅ β {π΅}) | |
9 | 8 | adantl 483 | . . 3 β’ ((π΄ β π β§ π΅ β π) β π΅ β {π΅}) |
10 | cnconst2 22657 | . . 3 β’ ((π« {π΄} β (TopOnβ{π΄}) β§ π« {π΅} β (TopOnβ{π΅}) β§ π΅ β {π΅}) β ({π΄} Γ {π΅}) β (π« {π΄} Cn π« {π΅})) | |
11 | 4, 7, 9, 10 | syl3anc 1372 | . 2 β’ ((π΄ β π β§ π΅ β π) β ({π΄} Γ {π΅}) β (π« {π΄} Cn π« {π΅})) |
12 | 1, 11 | eqeltrd 2834 | 1 β’ ((π΄ β π β§ π΅ β π) β (π₯ β {π΄} β¦ π΅) β (π« {π΄} Cn π« {π΅})) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β wcel 2107 Vcvv 3447 π« cpw 4564 {csn 4590 β¦ cmpt 5192 Γ cxp 5635 βcfv 6500 (class class class)co 7361 TopOnctopon 22282 Cn ccn 22598 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-1st 7925 df-2nd 7926 df-map 8773 df-topgen 17333 df-top 22266 df-topon 22283 df-cn 22601 df-cnp 22602 |
This theorem is referenced by: cncfdmsn 44221 |
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