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Mirrors > Home > MPE Home > Th. List > lubdm | Structured version Visualization version GIF version |
Description: Domain of the least upper bound function of a poset. (Contributed by NM, 6-Sep-2018.) |
Ref | Expression |
---|---|
lubfval.b | β’ π΅ = (BaseβπΎ) |
lubfval.l | β’ β€ = (leβπΎ) |
lubfval.u | β’ π = (lubβπΎ) |
lubfval.p | β’ (π β (βπ¦ β π π¦ β€ π₯ β§ βπ§ β π΅ (βπ¦ β π π¦ β€ π§ β π₯ β€ π§))) |
lubfval.k | β’ (π β πΎ β π) |
Ref | Expression |
---|---|
lubdm | β’ (π β dom π = {π β π« π΅ β£ β!π₯ β π΅ π}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lubfval.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
2 | lubfval.l | . . . 4 β’ β€ = (leβπΎ) | |
3 | lubfval.u | . . . 4 β’ π = (lubβπΎ) | |
4 | lubfval.p | . . . 4 β’ (π β (βπ¦ β π π¦ β€ π₯ β§ βπ§ β π΅ (βπ¦ β π π¦ β€ π§ β π₯ β€ π§))) | |
5 | lubfval.k | . . . 4 β’ (π β πΎ β π) | |
6 | 1, 2, 3, 4, 5 | lubfval 18165 | . . 3 β’ (π β π = ((π β π« π΅ β¦ (β©π₯ β π΅ π)) βΎ {π β£ β!π₯ β π΅ π})) |
7 | 6 | dmeqd 5847 | . 2 β’ (π β dom π = dom ((π β π« π΅ β¦ (β©π₯ β π΅ π)) βΎ {π β£ β!π₯ β π΅ π})) |
8 | riotaex 7297 | . . . . 5 β’ (β©π₯ β π΅ π) β V | |
9 | eqid 2736 | . . . . 5 β’ (π β π« π΅ β¦ (β©π₯ β π΅ π)) = (π β π« π΅ β¦ (β©π₯ β π΅ π)) | |
10 | 8, 9 | dmmpti 6628 | . . . 4 β’ dom (π β π« π΅ β¦ (β©π₯ β π΅ π)) = π« π΅ |
11 | 10 | ineq2i 4156 | . . 3 β’ ({π β£ β!π₯ β π΅ π} β© dom (π β π« π΅ β¦ (β©π₯ β π΅ π))) = ({π β£ β!π₯ β π΅ π} β© π« π΅) |
12 | dmres 5945 | . . 3 β’ dom ((π β π« π΅ β¦ (β©π₯ β π΅ π)) βΎ {π β£ β!π₯ β π΅ π}) = ({π β£ β!π₯ β π΅ π} β© dom (π β π« π΅ β¦ (β©π₯ β π΅ π))) | |
13 | dfrab2 4257 | . . 3 β’ {π β π« π΅ β£ β!π₯ β π΅ π} = ({π β£ β!π₯ β π΅ π} β© π« π΅) | |
14 | 11, 12, 13 | 3eqtr4i 2774 | . 2 β’ dom ((π β π« π΅ β¦ (β©π₯ β π΅ π)) βΎ {π β£ β!π₯ β π΅ π}) = {π β π« π΅ β£ β!π₯ β π΅ π} |
15 | 7, 14 | eqtrdi 2792 | 1 β’ (π β dom π = {π β π« π΅ β£ β!π₯ β π΅ π}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 396 = wceq 1540 β wcel 2105 {cab 2713 βwral 3061 β!wreu 3347 {crab 3403 β© cin 3897 π« cpw 4547 class class class wbr 5092 β¦ cmpt 5175 dom cdm 5620 βΎ cres 5622 βcfv 6479 β©crio 7292 Basecbs 17009 lecple 17066 lubclub 18124 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-lub 18161 |
This theorem is referenced by: lubeldm 18168 xrsclat 31576 isclatd 46628 |
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