Theorem spwpr4c 10009

Description: Supremum of an unordered pair of comparable elements.

Assertion

Ref	Expression
spwpr4c	|- ((R e. Poset /\ B e. U /\ ARB) -> (R supw {A, B}) = B)

Proof of Theorem spwpr4c

Step	Hyp	Ref	Expression
1		3simpa 872	. 2 |- ((R e. Poset /\ B e. U /\ ARB) -> (R e. Poset /\ B e. U))
2		simp3 878	. 2 |- ((R e. Poset /\ B e. U /\ ARB) -> ARB)
3		simp1 876	. . 3 |- ((R e. Poset /\ B e. U /\ ARB) -> R e. Poset)
4		brelrng 4190	. . . . 5 |- ((A e. _V /\ B e. U /\ ARB) -> B e. ran R)
5		brrelex 4028	. . . . . . 7 |- ((Rel R /\ ARB) -> A e. _V)
6		psrel 9989	. . . . . . 7 |- (R e. Poset -> Rel R)
7	5, 6	sylan 497	. . . . . 6 |- ((R e. Poset /\ ARB) -> A e. _V)
8	7	3adant2 895	. . . . 5 |- ((R e. Poset /\ B e. U /\ ARB) -> A e. _V)
9	4, 8	syld3an1 1143	. . . 4 |- ((R e. Poset /\ B e. U /\ ARB) -> B e. ran R)
10		eqid 1884	. . . . . 6 |- dom R = dom R
11	10	psrn 9993	. . . . 5 |- (R e. Poset -> dom R = ran R)
12	11	3ad2ant1 897	. . . 4 |- ((R e. Poset /\ B e. U /\ ARB) -> dom R = ran R)
13	9, 12	eleqtrrd 1974	. . 3 |- ((R e. Poset /\ B e. U /\ ARB) -> B e. dom R)
14	10	psref 9992	. . 3 |- ((R e. Poset /\ B e. dom R) -> BRB)
15	3, 13, 14	syl11anc 524	. 2 |- ((R e. Poset /\ B e. U /\ ARB) -> BRB)
16		simpr 350	. . . . 5 |- ((ARx /\ BRx) -> BRx)
17	16	a1i 8	. . . 4 |- (x e. dom R -> ((ARx /\ BRx) -> BRx))
18	17	rgen 2159	. . 3 |- A.x e. dom R((ARx /\ BRx) -> BRx)
19	18	a1i 8	. 2 |- ((R e. Poset /\ B e. U /\ ARB) -> A.x e. dom R((ARx /\ BRx) -> BRx))
20	10	spwpr4 10006	. 2 |- (((R e. Poset /\ B e. U) /\ (ARB /\ BRB) /\ A.x e. dom R((ARx /\ BRx) -> BRx)) -> (R supw {A, B}) = B)
21	1, 2, 15, 19, 20	syl121anc 1105	1 |- ((R e. Poset /\ B e. U /\ ARB) -> (R supw {A, B}) = B)

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292  {cpr 3045   class class class wbr 3338  dom cdm 3986  ran crn 3987  Rel wrel 3991  (class class class)co 4884  Posetcps 9980   sup_w cspw 9981

This theorem is referenced by:  nfwpr4c 14630  tolat 14631

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-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-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-fv 4014  df-opr 4886  df-oprab 4887  df-ps 9984  df-spw 9985

Copyright terms: Public domain