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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 18335 | . . 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 5902 | . 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 7374 | . . . . 5 β’ (β©π₯ β π΅ π) β V | |
9 | eqid 2728 | . . . . 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 6693 | . . . 4 β’ dom (π β π« π΅ β¦ (β©π₯ β π΅ π)) = π« π΅ |
11 | 10 | ineq2i 4205 | . . 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 6001 | . . 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 4306 | . . 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 2766 | . 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 2784 | 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 395 = wceq 1534 β wcel 2099 {cab 2705 βwral 3057 β!wreu 3370 {crab 3428 β© cin 3944 π« cpw 4598 class class class wbr 5142 β¦ cmpt 5225 dom cdm 5672 βΎ cres 5674 βcfv 6542 β©crio 7369 Basecbs 17173 lecple 17233 lubclub 18294 |
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 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 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-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-riota 7370 df-lub 18331 |
This theorem is referenced by: lubeldm 18338 xrsclat 32732 isclatd 47988 |
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