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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 18307 | . . 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 5896 | . 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 7362 | . . . . 5 β’ (β©π₯ β π΅ π) β V | |
9 | eqid 2724 | . . . . 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 6685 | . . . 4 β’ dom (π β π« π΅ β¦ (β©π₯ β π΅ π)) = π« π΅ |
11 | 10 | ineq2i 4202 | . . 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 5994 | . . 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 4303 | . . 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 2762 | . 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 2780 | 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 1533 β wcel 2098 {cab 2701 βwral 3053 β!wreu 3366 {crab 3424 β© cin 3940 π« cpw 4595 class class class wbr 5139 β¦ cmpt 5222 dom cdm 5667 βΎ cres 5669 βcfv 6534 β©crio 7357 Basecbs 17145 lecple 17205 lubclub 18266 |
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 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-lub 18303 |
This theorem is referenced by: lubeldm 18310 xrsclat 32651 isclatd 47820 |
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