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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 6720 | . . . 4 β’ (π β Fun πΉ) |
3 | 1 | frnd 6724 | . . . 4 β’ (π β ran πΉ β β) |
4 | fimacnvinrn2 7076 | . . . 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 583 | . . 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 44850 | . . . 4 β’ (π β ((-β[,)π΅) β© β) = (-β(,)π΅)) |
8 | 7 | imaeq2d 6057 | . . 3 β’ (π β (β‘πΉ β ((-β[,)π΅) β© β)) = (β‘πΉ β (-β(,)π΅))) |
9 | 5, 8 | eqtr2d 2768 | . 2 β’ (π β (β‘πΉ β (-β(,)π΅)) = (β‘πΉ β (-β[,)π΅))) |
10 | 1 | frexr 44680 | . . 3 β’ (π β πΉ:π΄βΆβ*) |
11 | 10, 6 | preimaicomnf 46012 | . 2 β’ (π β (β‘πΉ β (-β[,)π΅)) = {π₯ β π΄ β£ (πΉβπ₯) < π΅}) |
12 | 9, 11 | eqtrd 2767 | 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 1534 β wcel 2099 {crab 3427 β© cin 3943 β wss 3944 class class class wbr 5142 β‘ccnv 5671 ran crn 5673 β cima 5675 Fun wfun 6536 βΆwf 6538 βcfv 6542 (class class class)co 7414 βcr 11123 -βcmnf 11262 β*cxr 11263 < clt 11264 (,)cioo 13342 [,)cico 13344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-pre-lttri 11198 ax-pre-lttrn 11199 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7985 df-2nd 7986 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-ioo 13346 df-ico 13348 |
This theorem is referenced by: issmflem 46028 mbfresmf 46040 smfres 46091 smfco 46103 |
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