Theorem prsdm 28720

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 5036	. . . 4 |- dom .<_ = dom ( ( le ` K ) i^i ( B X. B ) )
3	2	eleq2i 2521	. . 3 |- ( x e. dom .<_ <-> x e. dom ( ( le ` K ) i^i ( B X. B ) ) )
4		ordtNEW.b	. . . . . . . . . 10 |- B = ( Base ` K )
5		eqid 2451	. . . . . . . . . 10 |- ( le ` K ) = ( le ` K )
6	4, 5	prsref 16177	. . . . . . . . 9 |- ( ( K e. Preset /\ x e. B ) -> x ( le ` K ) x )
7		df-br 4403	. . . . . . . . 9 |- ( x ( le ` K ) x <-> <. x , x >. e. ( le ` K ) )
8	6, 7	sylib 200	. . . . . . . 8 |- ( ( K e. Preset /\ x e. B ) -> <. x , x >. e. ( le ` K ) )
9		simpr 463	. . . . . . . . 9 |- ( ( K e. Preset /\ x e. B ) -> x e. B )
10		opelxpi 4866	. . . . . . . . 9 |- ( ( x e. B /\ x e. B ) -> <. x , x >. e. ( B X. B ) )
11	9, 10	sylancom 673	. . . . . . . 8 |- ( ( K e. Preset /\ x e. B ) -> <. x , x >. e. ( B X. B ) )
12	8, 11	elind 3618	. . . . . . 7 |- ( ( K e. Preset /\ x e. B ) -> <. x , x >. e. ( ( le ` K ) i^i ( B X. B ) ) )
13		vex 3048	. . . . . . . 8 |- x e. _V
14		opeq2 4167	. . . . . . . . 9 |- ( y = x -> <. x , y >. = <. x , x >. )
15	14	eleq1d 2513	. . . . . . . 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 3141	. . . . . . 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 17	. . . . . 6 |- ( ( K e. Preset /\ x e. B ) -> E. y <. x , y >. e. ( ( le ` K ) i^i ( B X. B ) ) )
18	17	ex 436	. . . . 5 |- ( K e. Preset -> ( x e. B -> E. y <. x , y >. e. ( ( le ` K ) i^i ( B X. B ) ) ) )
19		inss2 3653	. . . . . . . 8 |- ( ( le ` K ) i^i ( B X. B ) ) C_ ( B X. B )
20	19	sseli 3428	. . . . . . 7 |- ( <. x , y >. e. ( ( le ` K ) i^i ( B X. B ) ) -> <. x , y >. e. ( B X. B ) )
21		opelxp1 4867	. . . . . . 7 |- ( <. x , y >. e. ( B X. B ) -> x e. B )
22	20, 21	syl 17	. . . . . 6 |- ( <. x , y >. e. ( ( le ` K ) i^i ( B X. B ) ) -> x e. B )
23	22	exlimiv 1776	. . . . 5 |- ( E. y <. x , y >. e. ( ( le ` K ) i^i ( B X. B ) ) -> x e. B )
24	18, 23	impbid1 207	. . . 4 |- ( K e. Preset -> ( x e. B <-> E. y <. x , y >. e. ( ( le ` K ) i^i ( B X. B ) ) ) )
25	13	eldm2 5033	. . . 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 268	. . 3 |- ( K e. Preset -> ( x e. dom ( ( le ` K ) i^i ( B X. B ) ) <-> x e. B ) )
27	3, 26	syl5bb 261	. 2 |- ( K e. Preset -> ( x e. dom .<_ <-> x e. B ) )
28	27	eqrdv 2449	1 |- ( K e. Preset -> dom .<_ = B )

Syntax hints:   -> wi 4   /\ wa 371   = wceq 1444   E.wex 1663   e. wcel 1887   i^i cin 3403   <.cop 3974   class class class wbr 4402   X. cxp 4832   dom cdm 4834   ` cfv 5582   Basecbs 15121   lecple 15197   Preset cpreset 16171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-xp 4840  df-dm 4844  df-iota 5546  df-fv 5590  df-preset 16173

This theorem is referenced by:  prsssdm  28723  ordtprsval  28724  ordtprsuni  28725  ordtrestNEW  28727  ordtconlem1  28730