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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 4069 | . . 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 3995 | . . . . 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 7089 | . . . . . . 7 β’ ((π β§ π¦ β π΄) β (πΉβπ¦) β β) |
7 | 6 | mnfltd 13136 | . . . . . 6 β’ ((π β§ π¦ β π΄) β -β < (πΉβπ¦)) |
8 | 7 | ralrimiva 3136 | . . . . 5 β’ (π β βπ¦ β π΄ -β < (πΉβπ¦)) |
9 | nfcv 2892 | . . . . . . 7 β’ β²π₯-β | |
10 | nfcv 2892 | . . . . . . 7 β’ β²π₯ < | |
11 | pimgtmnf2.1 | . . . . . . . 8 β’ β²π₯πΉ | |
12 | nfcv 2892 | . . . . . . . 8 β’ β²π₯π¦ | |
13 | 11, 12 | nffv 6902 | . . . . . . 7 β’ β²π₯(πΉβπ¦) |
14 | 9, 10, 13 | nfbr 5190 | . . . . . 6 β’ β²π₯-β < (πΉβπ¦) |
15 | nfv 1909 | . . . . . 6 β’ β²π¦-β < (πΉβπ₯) | |
16 | fveq2 6892 | . . . . . . 7 β’ (π¦ = π₯ β (πΉβπ¦) = (πΉβπ₯)) | |
17 | 16 | breq2d 5155 | . . . . . 6 β’ (π¦ = π₯ β (-β < (πΉβπ¦) β -β < (πΉβπ₯))) |
18 | 14, 15, 17 | cbvralw 3294 | . . . . 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 510 | . . 3 β’ (π β (π΄ β π΄ β§ βπ₯ β π΄ -β < (πΉβπ₯))) |
21 | nfcv 2892 | . . . 4 β’ β²π₯π΄ | |
22 | 21, 21 | ssrabf 44545 | . . 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 3990 | 1 β’ (π β {π₯ β π΄ β£ -β < (πΉβπ₯)} = π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β²wnfc 2875 βwral 3051 {crab 3419 β wss 3939 class class class wbr 5143 βΆwf 6539 βcfv 6543 βcr 11137 -βcmnf 11276 < clt 11278 |
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 7738 ax-cnex 11194 |
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-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-ss 3956 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 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 |
This theorem is referenced by: (None) |
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