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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 2904 | . . 3 β’ β²π¦π΄ | |
3 | nfv 1918 | . . 3 β’ β²π¦(πΉβπ₯) < -β | |
4 | pimltmnf2f.1 | . . . . 5 β’ β²π₯πΉ | |
5 | nfcv 2904 | . . . . 5 β’ β²π₯π¦ | |
6 | 4, 5 | nffv 6902 | . . . 4 β’ β²π₯(πΉβπ¦) |
7 | nfcv 2904 | . . . 4 β’ β²π₯ < | |
8 | nfcv 2904 | . . . 4 β’ β²π₯-β | |
9 | 6, 7, 8 | nfbr 5196 | . . 3 β’ β²π₯(πΉβπ¦) < -β |
10 | fveq2 6892 | . . . 4 β’ (π₯ = π¦ β (πΉβπ₯) = (πΉβπ¦)) | |
11 | 10 | breq1d 5159 | . . 3 β’ (π₯ = π¦ β ((πΉβπ₯) < -β β (πΉβπ¦) < -β)) |
12 | 1, 2, 3, 9, 11 | cbvrabw 3468 | . 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 7087 | . . . . . . 7 β’ ((π β§ π¦ β π΄) β (πΉβπ¦) β β) |
15 | 14 | rexrd 11264 | . . . . . 6 β’ ((π β§ π¦ β π΄) β (πΉβπ¦) β β*) |
16 | 15 | mnfled 13115 | . . . . 5 β’ ((π β§ π¦ β π΄) β -β β€ (πΉβπ¦)) |
17 | mnfxr 11271 | . . . . . . 7 β’ -β β β* | |
18 | 17 | a1i 11 | . . . . . 6 β’ ((π β§ π¦ β π΄) β -β β β*) |
19 | 18, 15 | xrlenltd 11280 | . . . . 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 3147 | . . 3 β’ (π β βπ¦ β π΄ Β¬ (πΉβπ¦) < -β) |
22 | rabeq0 4385 | . . 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 2785 | 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 397 = wceq 1542 β wcel 2107 β²wnfc 2884 βwral 3062 {crab 3433 β c0 4323 class class class wbr 5149 βΆwf 6540 βcfv 6544 βcr 11109 -βcmnf 11246 β*cxr 11247 < clt 11248 β€ cle 11249 |
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 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 |
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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 |
This theorem is referenced by: pimltmnf2 45414 smfpimltxr 45463 |
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