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Mirrors > Home > MPE Home > Th. List > meetfval2 | Structured version Visualization version GIF version |
Description: Value of meet function for a poset. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.) |
Ref | Expression |
---|---|
meetfval.u | β’ πΊ = (glbβπΎ) |
meetfval.m | β’ β§ = (meetβπΎ) |
Ref | Expression |
---|---|
meetfval2 | β’ (πΎ β π β β§ = {β¨β¨π₯, π¦β©, π§β© β£ ({π₯, π¦} β dom πΊ β§ π§ = (πΊβ{π₯, π¦}))}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | meetfval.u | . . 3 β’ πΊ = (glbβπΎ) | |
2 | meetfval.m | . . 3 β’ β§ = (meetβπΎ) | |
3 | 1, 2 | meetfval 18341 | . 2 β’ (πΎ β π β β§ = {β¨β¨π₯, π¦β©, π§β© β£ {π₯, π¦}πΊπ§}) |
4 | 1 | glbfun 18319 | . . . . 5 β’ Fun πΊ |
5 | funbrfv2b 6939 | . . . . 5 β’ (Fun πΊ β ({π₯, π¦}πΊπ§ β ({π₯, π¦} β dom πΊ β§ (πΊβ{π₯, π¦}) = π§))) | |
6 | 4, 5 | ax-mp 5 | . . . 4 β’ ({π₯, π¦}πΊπ§ β ({π₯, π¦} β dom πΊ β§ (πΊβ{π₯, π¦}) = π§)) |
7 | eqcom 2731 | . . . . 5 β’ ((πΊβ{π₯, π¦}) = π§ β π§ = (πΊβ{π₯, π¦})) | |
8 | 7 | anbi2i 622 | . . . 4 β’ (({π₯, π¦} β dom πΊ β§ (πΊβ{π₯, π¦}) = π§) β ({π₯, π¦} β dom πΊ β§ π§ = (πΊβ{π₯, π¦}))) |
9 | 6, 8 | bitri 275 | . . 3 β’ ({π₯, π¦}πΊπ§ β ({π₯, π¦} β dom πΊ β§ π§ = (πΊβ{π₯, π¦}))) |
10 | 9 | oprabbii 7468 | . 2 β’ {β¨β¨π₯, π¦β©, π§β© β£ {π₯, π¦}πΊπ§} = {β¨β¨π₯, π¦β©, π§β© β£ ({π₯, π¦} β dom πΊ β§ π§ = (πΊβ{π₯, π¦}))} |
11 | 3, 10 | eqtrdi 2780 | 1 β’ (πΎ β π β β§ = {β¨β¨π₯, π¦β©, π§β© β£ ({π₯, π¦} β dom πΊ β§ π§ = (πΊβ{π₯, π¦}))}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 {cpr 4622 class class class wbr 5138 dom cdm 5666 Fun wfun 6527 βcfv 6533 {coprab 7402 glbcglb 18264 meetcmee 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 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
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 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-oprab 7405 df-glb 18301 df-meet 18303 |
This theorem is referenced by: meetdm 18343 meetval 18345 |
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