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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pimgtpnf2f | Structured version Visualization version GIF version |
Description: Given a real-valued function, the preimage of an open interval, unbounded above, with lower bound +β, is the empty set. (Contributed by Glauco Siliprandi, 15-Dec-2021.) |
Ref | Expression |
---|---|
pimgtpnf2f.1 | β’ β²π₯πΉ |
pimgtpnf2f.2 | β’ β²π₯π΄ |
pimgtpnf2f.3 | β’ (π β πΉ:π΄βΆβ) |
Ref | Expression |
---|---|
pimgtpnf2f | β’ (π β {π₯ β π΄ β£ +β < (πΉβπ₯)} = β ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pimgtpnf2f.2 | . . 3 β’ β²π₯π΄ | |
2 | nfcv 2892 | . . 3 β’ β²π¦π΄ | |
3 | nfv 1909 | . . 3 β’ β²π¦+β < (πΉβπ₯) | |
4 | nfcv 2892 | . . . 4 β’ β²π₯+β | |
5 | nfcv 2892 | . . . 4 β’ β²π₯ < | |
6 | pimgtpnf2f.1 | . . . . 5 β’ β²π₯πΉ | |
7 | nfcv 2892 | . . . . 5 β’ β²π₯π¦ | |
8 | 6, 7 | nffv 6901 | . . . 4 β’ β²π₯(πΉβπ¦) |
9 | 4, 5, 8 | nfbr 5190 | . . 3 β’ β²π₯+β < (πΉβπ¦) |
10 | fveq2 6891 | . . . 4 β’ (π₯ = π¦ β (πΉβπ₯) = (πΉβπ¦)) | |
11 | 10 | breq2d 5155 | . . 3 β’ (π₯ = π¦ β (+β < (πΉβπ₯) β +β < (πΉβπ¦))) |
12 | 1, 2, 3, 9, 11 | cbvrabw 3456 | . 2 β’ {π₯ β π΄ β£ +β < (πΉβπ₯)} = {π¦ β π΄ β£ +β < (πΉβπ¦)} |
13 | pimgtpnf2f.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 | pnfged 44918 | . . . . 5 β’ ((π β§ π¦ β π΄) β (πΉβπ¦) β€ +β) |
17 | pnfxr 11296 | . . . . . . 7 β’ +β β β* | |
18 | 17 | a1i 11 | . . . . . 6 β’ ((π β§ π¦ β π΄) β +β β β*) |
19 | 15, 18 | xrlenltd 11308 | . . . . 5 β’ ((π β§ π¦ β π΄) β ((πΉβπ¦) β€ +β β Β¬ +β < (πΉβπ¦))) |
20 | 16, 19 | mpbid 231 | . . . 4 β’ ((π β§ π¦ β π΄) β Β¬ +β < (πΉβπ¦)) |
21 | 20 | ralrimiva 3136 | . . 3 β’ (π β βπ¦ β π΄ Β¬ +β < (πΉβπ¦)) |
22 | rabeq0 4380 | . . 3 β’ ({π¦ β π΄ β£ +β < (πΉβπ¦)} = β β βπ¦ β π΄ Β¬ +β < (πΉβπ¦)) | |
23 | 21, 22 | sylibr 233 | . 2 β’ (π β {π¦ β π΄ β£ +β < (πΉβπ¦)} = β ) |
24 | 12, 23 | eqtrid 2777 | 1 β’ (π β {π₯ β π΄ β£ +β < (πΉβπ₯)} = 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 +βcpnf 11273 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-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-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 |
This theorem is referenced by: pimgtpnf2 46156 smfpimgtxr 46230 |
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