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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pimgtmnf2 | 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 whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
pimgtmnf2.1 | β’ β²π₯πΉ |
pimgtmnf2.2 | β’ (π β πΉ:π΄βΆβ) |
Ref | Expression |
---|---|
pimgtmnf2 | β’ (π β {π₯ β π΄ β£ -β < (πΉβπ₯)} = π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 4038 | . . 3 β’ {π₯ β π΄ β£ -β < (πΉβπ₯)} β π΄ | |
2 | 1 | a1i 11 | . 2 β’ (π β {π₯ β π΄ β£ -β < (πΉβπ₯)} β π΄) |
3 | ssid 3967 | . . . . 5 β’ π΄ β π΄ | |
4 | 3 | a1i 11 | . . . 4 β’ (π β π΄ β π΄) |
5 | pimgtmnf2.2 | . . . . . . . 8 β’ (π β πΉ:π΄βΆβ) | |
6 | 5 | ffvelcdmda 7036 | . . . . . . 7 β’ ((π β§ π¦ β π΄) β (πΉβπ¦) β β) |
7 | 6 | mnfltd 13050 | . . . . . 6 β’ ((π β§ π¦ β π΄) β -β < (πΉβπ¦)) |
8 | 7 | ralrimiva 3140 | . . . . 5 β’ (π β βπ¦ β π΄ -β < (πΉβπ¦)) |
9 | nfcv 2904 | . . . . . . 7 β’ β²π₯-β | |
10 | nfcv 2904 | . . . . . . 7 β’ β²π₯ < | |
11 | pimgtmnf2.1 | . . . . . . . 8 β’ β²π₯πΉ | |
12 | nfcv 2904 | . . . . . . . 8 β’ β²π₯π¦ | |
13 | 11, 12 | nffv 6853 | . . . . . . 7 β’ β²π₯(πΉβπ¦) |
14 | 9, 10, 13 | nfbr 5153 | . . . . . 6 β’ β²π₯-β < (πΉβπ¦) |
15 | nfv 1918 | . . . . . 6 β’ β²π¦-β < (πΉβπ₯) | |
16 | fveq2 6843 | . . . . . . 7 β’ (π¦ = π₯ β (πΉβπ¦) = (πΉβπ₯)) | |
17 | 16 | breq2d 5118 | . . . . . 6 β’ (π¦ = π₯ β (-β < (πΉβπ¦) β -β < (πΉβπ₯))) |
18 | 14, 15, 17 | cbvralw 3288 | . . . . 5 β’ (βπ¦ β π΄ -β < (πΉβπ¦) β βπ₯ β π΄ -β < (πΉβπ₯)) |
19 | 8, 18 | sylib 217 | . . . 4 β’ (π β βπ₯ β π΄ -β < (πΉβπ₯)) |
20 | 4, 19 | jca 513 | . . 3 β’ (π β (π΄ β π΄ β§ βπ₯ β π΄ -β < (πΉβπ₯))) |
21 | nfcv 2904 | . . . 4 β’ β²π₯π΄ | |
22 | 21, 21 | ssrabf 43412 | . . 3 β’ (π΄ β {π₯ β π΄ β£ -β < (πΉβπ₯)} β (π΄ β π΄ β§ βπ₯ β π΄ -β < (πΉβπ₯))) |
23 | 20, 22 | sylibr 233 | . 2 β’ (π β π΄ β {π₯ β π΄ β£ -β < (πΉβπ₯)}) |
24 | 2, 23 | eqssd 3962 | 1 β’ (π β {π₯ β π΄ β£ -β < (πΉβπ₯)} = π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β²wnfc 2884 βwral 3061 {crab 3406 β wss 3911 class class class wbr 5106 βΆwf 6493 βcfv 6497 βcr 11055 -βcmnf 11192 < clt 11194 |
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 |
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-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 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-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 |
This theorem is referenced by: (None) |
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