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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pimltmnf2f | Structured version Visualization version GIF version |
Description: Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound -β, is the empty set. (Contributed by Glauco Siliprandi, 15-Dec-2024.) |
Ref | Expression |
---|---|
pimltmnf2f.1 | β’ β²π₯πΉ |
pimltmnf2f.2 | β’ β²π₯π΄ |
pimltmnf2f.3 | β’ (π β πΉ:π΄βΆβ) |
Ref | Expression |
---|---|
pimltmnf2f | β’ (π β {π₯ β π΄ β£ (πΉβπ₯) < -β} = β ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pimltmnf2f.2 | . . 3 β’ β²π₯π΄ | |
2 | nfcv 2897 | . . 3 β’ β²π¦π΄ | |
3 | nfv 1909 | . . 3 β’ β²π¦(πΉβπ₯) < -β | |
4 | pimltmnf2f.1 | . . . . 5 β’ β²π₯πΉ | |
5 | nfcv 2897 | . . . . 5 β’ β²π₯π¦ | |
6 | 4, 5 | nffv 6895 | . . . 4 β’ β²π₯(πΉβπ¦) |
7 | nfcv 2897 | . . . 4 β’ β²π₯ < | |
8 | nfcv 2897 | . . . 4 β’ β²π₯-β | |
9 | 6, 7, 8 | nfbr 5188 | . . 3 β’ β²π₯(πΉβπ¦) < -β |
10 | fveq2 6885 | . . . 4 β’ (π₯ = π¦ β (πΉβπ₯) = (πΉβπ¦)) | |
11 | 10 | breq1d 5151 | . . 3 β’ (π₯ = π¦ β ((πΉβπ₯) < -β β (πΉβπ¦) < -β)) |
12 | 1, 2, 3, 9, 11 | cbvrabw 3461 | . 2 β’ {π₯ β π΄ β£ (πΉβπ₯) < -β} = {π¦ β π΄ β£ (πΉβπ¦) < -β} |
13 | pimltmnf2f.3 | . . . . . . . 8 β’ (π β πΉ:π΄βΆβ) | |
14 | 13 | ffvelcdmda 7080 | . . . . . . 7 β’ ((π β§ π¦ β π΄) β (πΉβπ¦) β β) |
15 | 14 | rexrd 11268 | . . . . . 6 β’ ((π β§ π¦ β π΄) β (πΉβπ¦) β β*) |
16 | 15 | mnfled 13121 | . . . . 5 β’ ((π β§ π¦ β π΄) β -β β€ (πΉβπ¦)) |
17 | mnfxr 11275 | . . . . . . 7 β’ -β β β* | |
18 | 17 | a1i 11 | . . . . . 6 β’ ((π β§ π¦ β π΄) β -β β β*) |
19 | 18, 15 | xrlenltd 11284 | . . . . 5 β’ ((π β§ π¦ β π΄) β (-β β€ (πΉβπ¦) β Β¬ (πΉβπ¦) < -β)) |
20 | 16, 19 | mpbid 231 | . . . 4 β’ ((π β§ π¦ β π΄) β Β¬ (πΉβπ¦) < -β) |
21 | 20 | ralrimiva 3140 | . . 3 β’ (π β βπ¦ β π΄ Β¬ (πΉβπ¦) < -β) |
22 | rabeq0 4379 | . . 3 β’ ({π¦ β π΄ β£ (πΉβπ¦) < -β} = β β βπ¦ β π΄ Β¬ (πΉβπ¦) < -β) | |
23 | 21, 22 | sylibr 233 | . 2 β’ (π β {π¦ β π΄ β£ (πΉβπ¦) < -β} = β ) |
24 | 12, 23 | eqtrid 2778 | 1 β’ (π β {π₯ β π΄ β£ (πΉβπ₯) < -β} = β ) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β²wnfc 2877 βwral 3055 {crab 3426 β c0 4317 class class class wbr 5141 βΆwf 6533 βcfv 6537 βcr 11111 -βcmnf 11250 β*cxr 11251 < clt 11252 β€ cle 11253 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 |
This theorem is referenced by: pimltmnf2 45991 smfpimltxr 46040 |
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