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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > preimaioomnf | Structured version Visualization version GIF version |
Description: Preimage of an open interval, unbounded below. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
preimaioomnf.1 | β’ (π β πΉ:π΄βΆβ) |
preimaioomnf.2 | β’ (π β π΅ β β*) |
Ref | Expression |
---|---|
preimaioomnf | β’ (π β (β‘πΉ β (-β(,)π΅)) = {π₯ β π΄ β£ (πΉβπ₯) < π΅}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | preimaioomnf.1 | . . . . 5 β’ (π β πΉ:π΄βΆβ) | |
2 | 1 | ffund 6722 | . . . 4 β’ (π β Fun πΉ) |
3 | 1 | frnd 6726 | . . . 4 β’ (π β ran πΉ β β) |
4 | fimacnvinrn2 7075 | . . . 4 β’ ((Fun πΉ β§ ran πΉ β β) β (β‘πΉ β (-β[,)π΅)) = (β‘πΉ β ((-β[,)π΅) β© β))) | |
5 | 2, 3, 4 | syl2anc 585 | . . 3 β’ (π β (β‘πΉ β (-β[,)π΅)) = (β‘πΉ β ((-β[,)π΅) β© β))) |
6 | preimaioomnf.2 | . . . . 5 β’ (π β π΅ β β*) | |
7 | 6 | icomnfinre 44265 | . . . 4 β’ (π β ((-β[,)π΅) β© β) = (-β(,)π΅)) |
8 | 7 | imaeq2d 6060 | . . 3 β’ (π β (β‘πΉ β ((-β[,)π΅) β© β)) = (β‘πΉ β (-β(,)π΅))) |
9 | 5, 8 | eqtr2d 2774 | . 2 β’ (π β (β‘πΉ β (-β(,)π΅)) = (β‘πΉ β (-β[,)π΅))) |
10 | 1 | frexr 44095 | . . 3 β’ (π β πΉ:π΄βΆβ*) |
11 | 10, 6 | preimaicomnf 45427 | . 2 β’ (π β (β‘πΉ β (-β[,)π΅)) = {π₯ β π΄ β£ (πΉβπ₯) < π΅}) |
12 | 9, 11 | eqtrd 2773 | 1 β’ (π β (β‘πΉ β (-β(,)π΅)) = {π₯ β π΄ β£ (πΉβπ₯) < π΅}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 {crab 3433 β© cin 3948 β wss 3949 class class class wbr 5149 β‘ccnv 5676 ran crn 5678 β cima 5680 Fun wfun 6538 βΆwf 6540 βcfv 6544 (class class class)co 7409 βcr 11109 -βcmnf 11246 β*cxr 11247 < clt 11248 (,)cioo 13324 [,)cico 13326 |
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 ax-resscn 11167 ax-pre-lttri 11184 ax-pre-lttrn 11185 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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-nel 3048 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 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-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-ioo 13328 df-ico 13330 |
This theorem is referenced by: issmflem 45443 mbfresmf 45455 smfres 45506 smfco 45518 |
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