Theorem suplub2 14616

Description: If it exists, a supremum of A is the least upper bound for A.

Hypothesis

Ref	Expression
suplub2.1	|- X = dom R

Assertion

Ref	Expression
suplub2	|- ((R e. Poset /\ A e. W /\ (R supw A) e. X) -> A.x e. (R ub A)(R supw A)Rx)

Distinct variable groups:   x,A   x,R   x,W   x,X

Proof of Theorem suplub2

Step	Hyp	Ref	Expression
1		eqid 1884	. . . . . . 7 |- dom R = dom R
2	1	puub2 14600	. . . . . 6 |- ((R e. Preset /\ A e. W) -> (x e. (R ub A) <-> (x e. dom R /\ A.z e. A zRx)))
3		posispre 14582	. . . . . 6 |- (R e. Poset -> R e. Preset )
4	2, 3	sylan 497	. . . . 5 |- ((R e. Poset /\ A e. W) -> (x e. (R ub A) <-> (x e. dom R /\ A.z e. A zRx)))
5	4	3adant3 896	. . . 4 |- ((R e. Poset /\ A e. W /\ (R supw A) e. X) -> (x e. (R ub A) <-> (x e. dom R /\ A.z e. A zRx)))
6	1	supdef 14604	. . . . . . . . . 10 |- ((R e. Poset /\ A e. W /\ (R supw A) e. dom R) -> (A.x e. A xR(R supw A) /\ A.x e. dom R(A.z e. A zRx -> (R supw A)Rx)))
7		suplub2.1	. . . . . . . . . . 11 |- X = dom R
8	7	eleq2i 1961	. . . . . . . . . 10 |- ((R supw A) e. X <-> (R supw A) e. dom R)
9	6, 8	syl3an3b 1135	. . . . . . . . 9 |- ((R e. Poset /\ A e. W /\ (R supw A) e. X) -> (A.x e. A xR(R supw A) /\ A.x e. dom R(A.z e. A zRx -> (R supw A)Rx)))
10	9	simprd 352	. . . . . . . 8 |- ((R e. Poset /\ A e. W /\ (R supw A) e. X) -> A.x e. dom R(A.z e. A zRx -> (R supw A)Rx))
11	10	r19.21bi 2188	. . . . . . 7 |- (((R e. Poset /\ A e. W /\ (R supw A) e. X) /\ x e. dom R) -> (A.z e. A zRx -> (R supw A)Rx))
12	11	ex 402	. . . . . 6 |- ((R e. Poset /\ A e. W /\ (R supw A) e. X) -> (x e. dom R -> (A.z e. A zRx -> (R supw A)Rx)))
13	12	com3l 38	. . . . 5 |- (x e. dom R -> (A.z e. A zRx -> ((R e. Poset /\ A e. W /\ (R supw A) e. X) -> (R supw A)Rx)))
14	13	imp 377	. . . 4 |- ((x e. dom R /\ A.z e. A zRx) -> ((R e. Poset /\ A e. W /\ (R supw A) e. X) -> (R supw A)Rx))
15	5, 14	syl6bi 231	. . 3 |- ((R e. Poset /\ A e. W /\ (R supw A) e. X) -> (x e. (R ub A) -> ((R e. Poset /\ A e. W /\ (R supw A) e. X) -> (R supw A)Rx)))
16	15	pm2.43a 80	. 2 |- ((R e. Poset /\ A e. W /\ (R supw A) e. X) -> (x e. (R ub A) -> (R supw A)Rx))
17	16	r19.21aiv 2175	1 |- ((R e. Poset /\ A e. W /\ (R supw A) e. X) -> A.x e. (R ub A)(R supw A)Rx)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105   class class class wbr 3338  dom cdm 3986  (class class class)co 4884  Posetcps 9980   sup_w cspw 9981   Preset cpreset 14555   ub cub 14558

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-ps 9984  df-spw 9985  df-prs 14563  df-ub 14596

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