![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
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 2892 | . . 3 β’ β²π¦π΄ | |
3 | nfv 1909 | . . 3 β’ β²π¦(πΉβπ₯) < -β | |
4 | pimltmnf2f.1 | . . . . 5 β’ β²π₯πΉ | |
5 | nfcv 2892 | . . . . 5 β’ β²π₯π¦ | |
6 | 4, 5 | nffv 6901 | . . . 4 β’ β²π₯(πΉβπ¦) |
7 | nfcv 2892 | . . . 4 β’ β²π₯ < | |
8 | nfcv 2892 | . . . 4 β’ β²π₯-β | |
9 | 6, 7, 8 | nfbr 5190 | . . 3 β’ β²π₯(πΉβπ¦) < -β |
10 | fveq2 6891 | . . . 4 β’ (π₯ = π¦ β (πΉβπ₯) = (πΉβπ¦)) | |
11 | 10 | breq1d 5153 | . . 3 β’ (π₯ = π¦ β ((πΉβπ₯) < -β β (πΉβπ¦) < -β)) |
12 | 1, 2, 3, 9, 11 | cbvrabw 3456 | . 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 7088 | . . . . . . 7 β’ ((π β§ π¦ β π΄) β (πΉβπ¦) β β) |
15 | 14 | rexrd 11292 | . . . . . 6 β’ ((π β§ π¦ β π΄) β (πΉβπ¦) β β*) |
16 | 15 | mnfled 13145 | . . . . 5 β’ ((π β§ π¦ β π΄) β -β β€ (πΉβπ¦)) |
17 | mnfxr 11299 | . . . . . . 7 β’ -β β β* | |
18 | 17 | a1i 11 | . . . . . 6 β’ ((π β§ π¦ β π΄) β -β β β*) |
19 | 18, 15 | xrlenltd 11308 | . . . . 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 3136 | . . 3 β’ (π β βπ¦ β π΄ Β¬ (πΉβπ¦) < -β) |
22 | rabeq0 4380 | . . 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 2777 | 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 394 = wceq 1533 β wcel 2098 β²wnfc 2875 βwral 3051 {crab 3419 β c0 4318 class class class wbr 5143 βΆwf 6538 βcfv 6542 βcr 11135 -βcmnf 11274 β*cxr 11275 < clt 11276 β€ cle 11277 |
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 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 |
This theorem is referenced by: pimltmnf2 46148 smfpimltxr 46197 |
Copyright terms: Public domain | W3C validator |