Theorem prsdm 28144

Description: Domain of the relation of a preset. (Contributed by Thierry Arnoux, 11-Sep-2015.)

Hypotheses

Ref	Expression
ordtNEW.b	|- B = ( Base ` K )
ordtNEW.l	|- .<_ = ( ( le ` K ) i^i ( B X. B ) )

Assertion

Ref	Expression
prsdm	|- ( K e. Preset -> dom .<_ = B )

Proof of Theorem prsdm

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordtNEW.l	. . . . 5 |- .<_ = ( ( le ` K ) i^i ( B X. B ) )
2	1	dmeqi 5214	. . . 4 |- dom .<_ = dom ( ( le ` K ) i^i ( B X. B ) )
3	2	eleq2i 2535	. . 3 |- ( x e. dom .<_ <-> x e. dom ( ( le ` K ) i^i ( B X. B ) ) )
4		ordtNEW.b	. . . . . . . . . 10 |- B = ( Base ` K )
5		eqid 2457	. . . . . . . . . 10 |- ( le ` K ) = ( le ` K )
6	4, 5	prsref 15779	. . . . . . . . 9 |- ( ( K e. Preset /\ x e. B ) -> x ( le ` K ) x )
7		df-br 4457	. . . . . . . . 9 |- ( x ( le ` K ) x <-> <. x , x >. e. ( le ` K ) )
8	6, 7	sylib 196	. . . . . . . 8 |- ( ( K e. Preset /\ x e. B ) -> <. x , x >. e. ( le ` K ) )
9		simpr 461	. . . . . . . . 9 |- ( ( K e. Preset /\ x e. B ) -> x e. B )
10		opelxpi 5040	. . . . . . . . 9 |- ( ( x e. B /\ x e. B ) -> <. x , x >. e. ( B X. B ) )
11	9, 10	sylancom 667	. . . . . . . 8 |- ( ( K e. Preset /\ x e. B ) -> <. x , x >. e. ( B X. B ) )
12	8, 11	elind 3684	. . . . . . 7 |- ( ( K e. Preset /\ x e. B ) -> <. x , x >. e. ( ( le ` K ) i^i ( B X. B ) ) )
13		vex 3112	. . . . . . . 8 |- x e. _V
14		opeq2 4220	. . . . . . . . 9 |- ( y = x -> <. x , y >. = <. x , x >. )
15	14	eleq1d 2526	. . . . . . . 8 |- ( y = x -> ( <. x , y >. e. ( ( le ` K ) i^i ( B X. B ) ) <-> <. x , x >. e. ( ( le ` K ) i^i ( B X. B ) ) ) )
16	13, 15	spcev 3201	. . . . . . 7 |- ( <. x , x >. e. ( ( le ` K ) i^i ( B X. B ) ) -> E. y <. x , y >. e. ( ( le ` K ) i^i ( B X. B ) ) )
17	12, 16	syl 16	. . . . . 6 |- ( ( K e. Preset /\ x e. B ) -> E. y <. x , y >. e. ( ( le ` K ) i^i ( B X. B ) ) )
18	17	ex 434	. . . . 5 |- ( K e. Preset -> ( x e. B -> E. y <. x , y >. e. ( ( le ` K ) i^i ( B X. B ) ) ) )
19		inss2 3715	. . . . . . . 8 |- ( ( le ` K ) i^i ( B X. B ) ) C_ ( B X. B )
20	19	sseli 3495	. . . . . . 7 |- ( <. x , y >. e. ( ( le ` K ) i^i ( B X. B ) ) -> <. x , y >. e. ( B X. B ) )
21		opelxp1 5041	. . . . . . 7 |- ( <. x , y >. e. ( B X. B ) -> x e. B )
22	20, 21	syl 16	. . . . . 6 |- ( <. x , y >. e. ( ( le ` K ) i^i ( B X. B ) ) -> x e. B )
23	22	exlimiv 1723	. . . . 5 |- ( E. y <. x , y >. e. ( ( le ` K ) i^i ( B X. B ) ) -> x e. B )
24	18, 23	impbid1 203	. . . 4 |- ( K e. Preset -> ( x e. B <-> E. y <. x , y >. e. ( ( le ` K ) i^i ( B X. B ) ) ) )
25	13	eldm2 5211	. . . 4 |- ( x e. dom ( ( le ` K ) i^i ( B X. B ) ) <-> E. y <. x , y >. e. ( ( le ` K ) i^i ( B X. B ) ) )
26	24, 25	syl6rbbr 264	. . 3 |- ( K e. Preset -> ( x e. dom ( ( le ` K ) i^i ( B X. B ) ) <-> x e. B ) )
27	3, 26	syl5bb 257	. 2 |- ( K e. Preset -> ( x e. dom .<_ <-> x e. B ) )
28	27	eqrdv 2454	1 |- ( K e. Preset -> dom .<_ = B )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1395   E.wex 1613   e. wcel 1819   i^i cin 3470   <.cop 4038   class class class wbr 4456   X. cxp 5006   dom cdm 5008   ` cfv 5594   Basecbs 14735   lecple 14810   Preset cpreset 15773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-xp 5014  df-dm 5018  df-iota 5557  df-fv 5602  df-preset 15775

This theorem is referenced by:  prsssdm  28147  ordtprsval  28148  ordtprsuni  28149  ordtrestNEW  28151  ordtconlem1  28154