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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pimltpnf2 | 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 whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 15-Dec-2024.) |
Ref | Expression |
---|---|
pimltpnf2.1 | β’ β²π₯πΉ |
pimltpnf2.2 | β’ (π β πΉ:π΄βΆβ) |
Ref | Expression |
---|---|
pimltpnf2 | β’ (π β {π₯ β π΄ β£ (πΉβπ₯) < +β} = π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pimltpnf2.1 | . 2 β’ β²π₯πΉ | |
2 | nfcv 2904 | . 2 β’ β²π₯π΄ | |
3 | pimltpnf2.2 | . 2 β’ (π β πΉ:π΄βΆβ) | |
4 | 1, 2, 3 | pimltpnf2f 45428 | 1 β’ (π β {π₯ β π΄ β£ (πΉβπ₯) < +β} = π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β²wnfc 2884 {crab 3433 class class class wbr 5149 βΆwf 6540 βcfv 6544 βcr 11109 +βcpnf 11245 < clt 11248 |
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 |
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-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 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-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fv 6552 df-pnf 11250 df-xr 11252 df-ltxr 11253 |
This theorem is referenced by: (None) |
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