Theorem psdmrn 17207

Description: The domain and range of a poset equal its field. (Contributed by NM, 13-May-2008.)

Assertion

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

Proof of Theorem psdmrn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssun1 3776	. . . . 5 |- dom R C_ ( dom R u. ran R )
2		dmrnssfld 5384	. . . . 5 |- ( dom R u. ran R ) C_ U. U. R
3	1, 2	sstri 3612	. . . 4 |- dom R C_ U. U. R
4	3	a1i 11	. . 3 |- ( R e. PosetRel -> dom R C_ U. U. R )
5		pslem 17206	. . . . . 6 |- ( R e. PosetRel -> ( ( ( x R x /\ x R x ) -> x R x ) /\ ( x e. U. U. R -> x R x ) /\ ( ( x R x /\ x R x ) -> x = x ) ) )
6	5	simp2d 1074	. . . . 5 |- ( R e. PosetRel -> ( x e. U. U. R -> x R x ) )
7		vex 3203	. . . . . 6 |- x e. _V
8	7, 7	breldm 5329	. . . . 5 |- ( x R x -> x e. dom R )
9	6, 8	syl6 35	. . . 4 |- ( R e. PosetRel -> ( x e. U. U. R -> x e. dom R ) )
10	9	ssrdv 3609	. . 3 |- ( R e. PosetRel -> U. U. R C_ dom R )
11	4, 10	eqssd 3620	. 2 |- ( R e. PosetRel -> dom R = U. U. R )
12		ssun2 3777	. . . . 5 |- ran R C_ ( dom R u. ran R )
13	12, 2	sstri 3612	. . . 4 |- ran R C_ U. U. R
14	13	a1i 11	. . 3 |- ( R e. PosetRel -> ran R C_ U. U. R )
15	7, 7	brelrn 5356	. . . . 5 |- ( x R x -> x e. ran R )
16	6, 15	syl6 35	. . . 4 |- ( R e. PosetRel -> ( x e. U. U. R -> x e. ran R ) )
17	16	ssrdv 3609	. . 3 |- ( R e. PosetRel -> U. U. R C_ ran R )
18	14, 17	eqssd 3620	. 2 |- ( R e. PosetRel -> ran R = U. U. R )
19	11, 18	jca 554	1 |- ( R e. PosetRel -> ( dom R = U. U. R /\ ran R = U. U. R ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   u. cun 3572   C_ wss 3574   U.cuni 4436   class class class wbr 4653   dom cdm 5114   ran crn 5115   PosetRelcps 17198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ps 17200

This theorem is referenced by:  psref  17208  psrn  17209  psss  17214  tsrdir  17238