Theorem psdmrn 8644

Description: The domain and range of a poset equal its field.

Assertion

Ref	Expression
psdmrn	|- (R e. Poset -> (dom R = U.U.R /\ ran R = U.U.R))

Proof of Theorem psdmrn

Step	Hyp	Ref	Expression
1		ssun1 2196	. . . . 5 |- dom R (_ (dom R u. ran R)
2		dmrnssfld 3363	. . . . 5 |- (dom R u. ran R) (_ U.U.R
3	1, 2	sstri 2076	. . . 4 |- dom R (_ U.U.R
4	3	a1i 8	. . 3 |- (R e. Poset -> dom R (_ U.U.R)
5		pslem 8643	. . . . . . . . . 10 |- (R e. Poset -> ((x e. U.U.R /\ x e. U.U.R /\ x e. U.U.R) -> (((xRx /\ xRx) -> xRx) /\ ((x e. U.U.R -> xRx) /\ ((xRx /\ xRx) -> x = x)))))
6		simprl 416	. . . . . . . . . 10 |- ((((xRx /\ xRx) -> xRx) /\ ((x e. U.U.R -> xRx) /\ ((xRx /\ xRx) -> x = x))) -> (x e. U.U.R -> xRx))
7	5, 6	syl6 22	. . . . . . . . 9 |- (R e. Poset -> ((x e. U.U.R /\ x e. U.U.R /\ x e. U.U.R) -> (x e. U.U.R -> xRx)))
8	7	3expd 852	. . . . . . . 8 |- (R e. Poset -> (x e. U.U.R -> (x e. U.U.R -> (x e. U.U.R -> (x e. U.U.R -> xRx)))))
9	8	pm2.43d 65	. . . . . . 7 |- (R e. Poset -> (x e. U.U.R -> (x e. U.U.R -> (x e. U.U.R -> xRx))))
10	9	pm2.43d 65	. . . . . 6 |- (R e. Poset -> (x e. U.U.R -> (x e. U.U.R -> xRx)))
11	10	pm2.43d 65	. . . . 5 |- (R e. Poset -> (x e. U.U.R -> xRx))
12		visset 1816	. . . . . 6 |- x e. V
13	12	breldm 3321	. . . . 5 |- (xRx -> x e. dom R)
14	11, 13	syl6 22	. . . 4 |- (R e. Poset -> (x e. U.U.R -> x e. dom R))
15	14	ssrdv 2073	. . 3 |- (R e. Poset -> U.U.R (_ dom R)
16	4, 15	eqssd 2082	. 2 |- (R e. Poset -> dom R = U.U.R)
17		ssun2 2197	. . . . 5 |- ran R (_ (dom R u. ran R)
18	17, 2	sstri 2076	. . . 4 |- ran R (_ U.U.R
19	18	a1i 8	. . 3 |- (R e. Poset -> ran R (_ U.U.R)
20	12, 12	brelrn 3350	. . . . 5 |- (xRx -> x e. ran R)
21	11, 20	syl6 22	. . . 4 |- (R e. Poset -> (x e. U.U.R -> x e. ran R))
22	21	ssrdv 2073	. . 3 |- (R e. Poset -> U.U.R (_ ran R)
23	19, 22	eqssd 2082	. 2 |- (R e. Poset -> ran R = U.U.R)
24	16, 23	jca 288	1 |- (R e. Poset -> (dom R = U.U.R /\ ran R = U.U.R))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960   u. cun 2048   (_ wss 2050  U.cuni 2507   class class class wbr 2624  dom cdm 3176  ran crn 3177  Posetcps 8629

This theorem is referenced by:  psref 8645  psrn 8646  spwval 8655  spwnex 8657

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ps 8635

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