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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 6853 | . . . 4 β’ β²π₯(πΉβπ¦) |
7 | nfcv 2904 | . . . 4 β’ β²π₯ < | |
8 | nfcv 2904 | . . . 4 β’ β²π₯-β | |
9 | 6, 7, 8 | nfbr 5153 | . . 3 β’ β²π₯(πΉβπ¦) < -β |
10 | fveq2 6843 | . . . 4 β’ (π₯ = π¦ β (πΉβπ₯) = (πΉβπ¦)) | |
11 | 10 | breq1d 5116 | . . 3 β’ (π₯ = π¦ β ((πΉβπ₯) < -β β (πΉβπ¦) < -β)) |
12 | 1, 2, 3, 9, 11 | cbvrabw 3438 | . 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 7036 | . . . . . . 7 β’ ((π β§ π¦ β π΄) β (πΉβπ¦) β β) |
15 | 14 | rexrd 11210 | . . . . . 6 β’ ((π β§ π¦ β π΄) β (πΉβπ¦) β β*) |
16 | 15 | mnfled 13061 | . . . . 5 β’ ((π β§ π¦ β π΄) β -β β€ (πΉβπ¦)) |
17 | mnfxr 11217 | . . . . . . 7 β’ -β β β* | |
18 | 17 | a1i 11 | . . . . . 6 β’ ((π β§ π¦ β π΄) β -β β β*) |
19 | 18, 15 | xrlenltd 11226 | . . . . 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 4345 | . . 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 3061 {crab 3406 β c0 4283 class class class wbr 5106 βΆwf 6493 βcfv 6497 βcr 11055 -βcmnf 11192 β*cxr 11193 < clt 11194 β€ cle 11195 |
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 |
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 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-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 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-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 |
This theorem is referenced by: pimltmnf2 45025 smfpimltxr 45074 |
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