![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > fcnre | Structured version Visualization version GIF version |
Description: A function continuous with respect to the standard topology, is a real mapping. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Ref | Expression |
---|---|
fcnre.1 | β’ πΎ = (topGenβran (,)) |
fcnre.3 | β’ π = βͺ π½ |
sfcnre.5 | β’ πΆ = (π½ Cn πΎ) |
fcnre.6 | β’ (π β πΉ β πΆ) |
Ref | Expression |
---|---|
fcnre | β’ (π β πΉ:πβΆβ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fcnre.6 | . . . . 5 β’ (π β πΉ β πΆ) | |
2 | sfcnre.5 | . . . . 5 β’ πΆ = (π½ Cn πΎ) | |
3 | 1, 2 | eleqtrdi 2844 | . . . 4 β’ (π β πΉ β (π½ Cn πΎ)) |
4 | cntop1 22607 | . . . 4 β’ (πΉ β (π½ Cn πΎ) β π½ β Top) | |
5 | 3, 4 | syl 17 | . . 3 β’ (π β π½ β Top) |
6 | fcnre.3 | . . . 4 β’ π = βͺ π½ | |
7 | 6 | toptopon 22282 | . . 3 β’ (π½ β Top β π½ β (TopOnβπ)) |
8 | 5, 7 | sylib 217 | . 2 β’ (π β π½ β (TopOnβπ)) |
9 | fcnre.1 | . . . 4 β’ πΎ = (topGenβran (,)) | |
10 | retopon 24143 | . . . 4 β’ (topGenβran (,)) β (TopOnββ) | |
11 | 9, 10 | eqeltri 2830 | . . 3 β’ πΎ β (TopOnββ) |
12 | 11 | a1i 11 | . 2 β’ (π β πΎ β (TopOnββ)) |
13 | cnf2 22616 | . 2 β’ ((π½ β (TopOnβπ) β§ πΎ β (TopOnββ) β§ πΉ β (π½ Cn πΎ)) β πΉ:πβΆβ) | |
14 | 8, 12, 3, 13 | syl3anc 1372 | 1 β’ (π β πΉ:πβΆβ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 βͺ cuni 4866 ran crn 5635 βΆwf 6493 βcfv 6497 (class class class)co 7358 βcr 11055 (,)cioo 13270 topGenctg 17324 Topctop 22258 TopOnctopon 22275 Cn ccn 22591 |
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 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-pre-lttri 11130 ax-pre-lttrn 11131 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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-nel 3047 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-ioo 13274 df-topgen 17330 df-top 22259 df-topon 22276 df-bases 22312 df-cn 22594 |
This theorem is referenced by: rfcnpre2 43324 cncmpmax 43325 rfcnpre3 43326 rfcnpre4 43327 rfcnnnub 43329 stoweidlem28 44355 stoweidlem29 44356 stoweidlem36 44363 stoweidlem43 44370 stoweidlem44 44371 stoweidlem47 44374 stoweidlem52 44379 stoweidlem57 44384 stoweidlem59 44386 stoweidlem60 44387 stoweidlem61 44388 stoweidlem62 44389 stoweid 44390 |
Copyright terms: Public domain | W3C validator |