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Mirrors > Home > MPE Home > Th. List > joindef | Structured version Visualization version GIF version |
Description: Two ways to say that a join is defined. (Contributed by NM, 9-Sep-2018.) |
Ref | Expression |
---|---|
joindef.u | β’ π = (lubβπΎ) |
joindef.j | β’ β¨ = (joinβπΎ) |
joindef.k | β’ (π β πΎ β π) |
joindef.x | β’ (π β π β π) |
joindef.y | β’ (π β π β π) |
Ref | Expression |
---|---|
joindef | β’ (π β (β¨π, πβ© β dom β¨ β {π, π} β dom π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | joindef.k | . . 3 β’ (π β πΎ β π) | |
2 | joindef.u | . . . . 5 β’ π = (lubβπΎ) | |
3 | joindef.j | . . . . 5 β’ β¨ = (joinβπΎ) | |
4 | 2, 3 | joindm 18337 | . . . 4 β’ (πΎ β π β dom β¨ = {β¨π₯, π¦β© β£ {π₯, π¦} β dom π}) |
5 | 4 | eleq2d 2813 | . . 3 β’ (πΎ β π β (β¨π, πβ© β dom β¨ β β¨π, πβ© β {β¨π₯, π¦β© β£ {π₯, π¦} β dom π})) |
6 | 1, 5 | syl 17 | . 2 β’ (π β (β¨π, πβ© β dom β¨ β β¨π, πβ© β {β¨π₯, π¦β© β£ {π₯, π¦} β dom π})) |
7 | joindef.x | . . 3 β’ (π β π β π) | |
8 | joindef.y | . . 3 β’ (π β π β π) | |
9 | preq1 4732 | . . . . 5 β’ (π₯ = π β {π₯, π¦} = {π, π¦}) | |
10 | 9 | eleq1d 2812 | . . . 4 β’ (π₯ = π β ({π₯, π¦} β dom π β {π, π¦} β dom π)) |
11 | preq2 4733 | . . . . 5 β’ (π¦ = π β {π, π¦} = {π, π}) | |
12 | 11 | eleq1d 2812 | . . . 4 β’ (π¦ = π β ({π, π¦} β dom π β {π, π} β dom π)) |
13 | 10, 12 | opelopabg 5531 | . . 3 β’ ((π β π β§ π β π) β (β¨π, πβ© β {β¨π₯, π¦β© β£ {π₯, π¦} β dom π} β {π, π} β dom π)) |
14 | 7, 8, 13 | syl2anc 583 | . 2 β’ (π β (β¨π, πβ© β {β¨π₯, π¦β© β£ {π₯, π¦} β dom π} β {π, π} β dom π)) |
15 | 6, 14 | bitrd 279 | 1 β’ (π β (β¨π, πβ© β dom β¨ β {π, π} β dom π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1533 β wcel 2098 {cpr 4625 β¨cop 4629 {copab 5203 dom cdm 5669 βcfv 6536 lubclub 18271 joincjn 18273 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-oprab 7408 df-lub 18308 df-join 18310 |
This theorem is referenced by: joinval 18339 joincl 18340 joindmss 18341 joineu 18344 clatl 18470 joindm2 47857 |
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