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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rfcnpre2 | Structured version Visualization version GIF version |
Description: If πΉ is a continuous function with respect to the standard topology, then the preimage A of the values smaller than a given extended real π΅, is an open set. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Ref | Expression |
---|---|
rfcnpre2.1 | β’ β²π₯π΅ |
rfcnpre2.2 | β’ β²π₯πΉ |
rfcnpre2.3 | β’ β²π₯π |
rfcnpre2.4 | β’ πΎ = (topGenβran (,)) |
rfcnpre2.5 | β’ π = βͺ π½ |
rfcnpre2.6 | β’ π΄ = {π₯ β π β£ (πΉβπ₯) < π΅} |
rfcnpre2.7 | β’ (π β π΅ β β*) |
rfcnpre2.8 | β’ (π β πΉ β (π½ Cn πΎ)) |
Ref | Expression |
---|---|
rfcnpre2 | β’ (π β π΄ β π½) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rfcnpre2.3 | . . . 4 β’ β²π₯π | |
2 | rfcnpre2.2 | . . . . . 6 β’ β²π₯πΉ | |
3 | 2 | nfcnv 5878 | . . . . 5 β’ β²π₯β‘πΉ |
4 | nfcv 2902 | . . . . . 6 β’ β²π₯-β | |
5 | nfcv 2902 | . . . . . 6 β’ β²π₯(,) | |
6 | rfcnpre2.1 | . . . . . 6 β’ β²π₯π΅ | |
7 | 4, 5, 6 | nfov 7442 | . . . . 5 β’ β²π₯(-β(,)π΅) |
8 | 3, 7 | nfima 6067 | . . . 4 β’ β²π₯(β‘πΉ β (-β(,)π΅)) |
9 | nfrab1 3450 | . . . 4 β’ β²π₯{π₯ β π β£ (πΉβπ₯) < π΅} | |
10 | rfcnpre2.4 | . . . . . . . . 9 β’ πΎ = (topGenβran (,)) | |
11 | rfcnpre2.5 | . . . . . . . . 9 β’ π = βͺ π½ | |
12 | eqid 2731 | . . . . . . . . 9 β’ (π½ Cn πΎ) = (π½ Cn πΎ) | |
13 | rfcnpre2.8 | . . . . . . . . 9 β’ (π β πΉ β (π½ Cn πΎ)) | |
14 | 10, 11, 12, 13 | fcnre 44012 | . . . . . . . 8 β’ (π β πΉ:πβΆβ) |
15 | 14 | ffvelcdmda 7086 | . . . . . . 7 β’ ((π β§ π₯ β π) β (πΉβπ₯) β β) |
16 | rfcnpre2.7 | . . . . . . . . 9 β’ (π β π΅ β β*) | |
17 | elioomnf 13426 | . . . . . . . . 9 β’ (π΅ β β* β ((πΉβπ₯) β (-β(,)π΅) β ((πΉβπ₯) β β β§ (πΉβπ₯) < π΅))) | |
18 | 16, 17 | syl 17 | . . . . . . . 8 β’ (π β ((πΉβπ₯) β (-β(,)π΅) β ((πΉβπ₯) β β β§ (πΉβπ₯) < π΅))) |
19 | 18 | baibd 539 | . . . . . . 7 β’ ((π β§ (πΉβπ₯) β β) β ((πΉβπ₯) β (-β(,)π΅) β (πΉβπ₯) < π΅)) |
20 | 15, 19 | syldan 590 | . . . . . 6 β’ ((π β§ π₯ β π) β ((πΉβπ₯) β (-β(,)π΅) β (πΉβπ₯) < π΅)) |
21 | 20 | pm5.32da 578 | . . . . 5 β’ (π β ((π₯ β π β§ (πΉβπ₯) β (-β(,)π΅)) β (π₯ β π β§ (πΉβπ₯) < π΅))) |
22 | ffn 6717 | . . . . . 6 β’ (πΉ:πβΆβ β πΉ Fn π) | |
23 | elpreima 7059 | . . . . . 6 β’ (πΉ Fn π β (π₯ β (β‘πΉ β (-β(,)π΅)) β (π₯ β π β§ (πΉβπ₯) β (-β(,)π΅)))) | |
24 | 14, 22, 23 | 3syl 18 | . . . . 5 β’ (π β (π₯ β (β‘πΉ β (-β(,)π΅)) β (π₯ β π β§ (πΉβπ₯) β (-β(,)π΅)))) |
25 | rabid 3451 | . . . . . 6 β’ (π₯ β {π₯ β π β£ (πΉβπ₯) < π΅} β (π₯ β π β§ (πΉβπ₯) < π΅)) | |
26 | 25 | a1i 11 | . . . . 5 β’ (π β (π₯ β {π₯ β π β£ (πΉβπ₯) < π΅} β (π₯ β π β§ (πΉβπ₯) < π΅))) |
27 | 21, 24, 26 | 3bitr4d 311 | . . . 4 β’ (π β (π₯ β (β‘πΉ β (-β(,)π΅)) β π₯ β {π₯ β π β£ (πΉβπ₯) < π΅})) |
28 | 1, 8, 9, 27 | eqrd 4001 | . . 3 β’ (π β (β‘πΉ β (-β(,)π΅)) = {π₯ β π β£ (πΉβπ₯) < π΅}) |
29 | rfcnpre2.6 | . . 3 β’ π΄ = {π₯ β π β£ (πΉβπ₯) < π΅} | |
30 | 28, 29 | eqtr4di 2789 | . 2 β’ (π β (β‘πΉ β (-β(,)π΅)) = π΄) |
31 | iooretop 24503 | . . . . 5 β’ (-β(,)π΅) β (topGenβran (,)) | |
32 | 31 | a1i 11 | . . . 4 β’ (π β (-β(,)π΅) β (topGenβran (,))) |
33 | 32, 10 | eleqtrrdi 2843 | . . 3 β’ (π β (-β(,)π΅) β πΎ) |
34 | cnima 22990 | . . 3 β’ ((πΉ β (π½ Cn πΎ) β§ (-β(,)π΅) β πΎ) β (β‘πΉ β (-β(,)π΅)) β π½) | |
35 | 13, 33, 34 | syl2anc 583 | . 2 β’ (π β (β‘πΉ β (-β(,)π΅)) β π½) |
36 | 30, 35 | eqeltrrd 2833 | 1 β’ (π β π΄ β π½) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1540 β²wnf 1784 β wcel 2105 β²wnfc 2882 {crab 3431 βͺ cuni 4908 class class class wbr 5148 β‘ccnv 5675 ran crn 5677 β cima 5679 Fn wfn 6538 βΆwf 6539 βcfv 6543 (class class class)co 7412 βcr 11113 -βcmnf 11251 β*cxr 11252 < clt 11253 (,)cioo 13329 topGenctg 17388 Cn ccn 22949 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-er 8707 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-sup 9441 df-inf 9442 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-n0 12478 df-z 12564 df-uz 12828 df-q 12938 df-ioo 13333 df-topgen 17394 df-top 22617 df-topon 22634 df-bases 22670 df-cn 22952 |
This theorem is referenced by: stoweidlem52 45067 cnfsmf 45755 |
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