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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pimltpnf2f | 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, 15-Dec-2024.) |
Ref | Expression |
---|---|
pimltpnf2f.1 | β’ β²π₯πΉ |
pimltpnf2f.2 | β’ β²π₯π΄ |
pimltpnf2f.3 | β’ (π β πΉ:π΄βΆβ) |
Ref | Expression |
---|---|
pimltpnf2f | β’ (π β {π₯ β π΄ β£ (πΉβπ₯) < +β} = π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pimltpnf2f.2 | . . 3 β’ β²π₯π΄ | |
2 | nfcv 2892 | . . 3 β’ β²π¦π΄ | |
3 | nfv 1909 | . . 3 β’ β²π¦(πΉβπ₯) < +β | |
4 | pimltpnf2f.1 | . . . . 5 β’ β²π₯πΉ | |
5 | nfcv 2892 | . . . . 5 β’ β²π₯π¦ | |
6 | 4, 5 | nffv 6901 | . . . 4 β’ β²π₯(πΉβπ¦) |
7 | nfcv 2892 | . . . 4 β’ β²π₯ < | |
8 | nfcv 2892 | . . . 4 β’ β²π₯+β | |
9 | 6, 7, 8 | nfbr 5190 | . . 3 β’ β²π₯(πΉβπ¦) < +β |
10 | fveq2 6891 | . . . 4 β’ (π₯ = π¦ β (πΉβπ₯) = (πΉβπ¦)) | |
11 | 10 | breq1d 5153 | . . 3 β’ (π₯ = π¦ β ((πΉβπ₯) < +β β (πΉβπ¦) < +β)) |
12 | 1, 2, 3, 9, 11 | cbvrabw 3456 | . 2 β’ {π₯ β π΄ β£ (πΉβπ₯) < +β} = {π¦ β π΄ β£ (πΉβπ¦) < +β} |
13 | nfv 1909 | . . 3 β’ β²π¦π | |
14 | pimltpnf2f.3 | . . . 4 β’ (π β πΉ:π΄βΆβ) | |
15 | 14 | ffvelcdmda 7088 | . . 3 β’ ((π β§ π¦ β π΄) β (πΉβπ¦) β β) |
16 | 13, 15 | pimltpnf 46154 | . 2 β’ (π β {π¦ β π΄ β£ (πΉβπ¦) < +β} = π΄) |
17 | 12, 16 | eqtrid 2777 | 1 β’ (π β {π₯ β π΄ β£ (πΉβπ₯) < +β} = π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β²wnfc 2875 {crab 3419 class class class wbr 5143 βΆwf 6538 βcfv 6542 βcr 11135 +βcpnf 11273 < clt 11276 |
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 |
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 3943 df-un 3945 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-xr 11280 df-ltxr 11281 |
This theorem is referenced by: pimltpnf2 46163 smfpimltxr 46197 |
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