Theorem prsdm 28794

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 5041	. . . 4 |- dom .<_ = dom ( ( le ` K ) i^i ( B X. B ) )
3	2	eleq2i 2541	. . 3 |- ( x e. dom .<_ <-> x e. dom ( ( le ` K ) i^i ( B X. B ) ) )
4		ordtNEW.b	. . . . . . . . . 10 |- B = ( Base ` K )
5		eqid 2471	. . . . . . . . . 10 |- ( le ` K ) = ( le ` K )
6	4, 5	prsref 16255	. . . . . . . . 9 |- ( ( K e. Preset /\ x e. B ) -> x ( le ` K ) x )
7		df-br 4396	. . . . . . . . 9 |- ( x ( le ` K ) x <-> <. x , x >. e. ( le ` K ) )
8	6, 7	sylib 201	. . . . . . . 8 |- ( ( K e. Preset /\ x e. B ) -> <. x , x >. e. ( le ` K ) )
9		simpr 468	. . . . . . . . 9 |- ( ( K e. Preset /\ x e. B ) -> x e. B )
10		opelxpi 4871	. . . . . . . . 9 |- ( ( x e. B /\ x e. B ) -> <. x , x >. e. ( B X. B ) )
11	9, 10	sylancom 680	. . . . . . . 8 |- ( ( K e. Preset /\ x e. B ) -> <. x , x >. e. ( B X. B ) )
12	8, 11	elind 3609	. . . . . . 7 |- ( ( K e. Preset /\ x e. B ) -> <. x , x >. e. ( ( le ` K ) i^i ( B X. B ) ) )
13		vex 3034	. . . . . . . 8 |- x e. _V
14		opeq2 4159	. . . . . . . . 9 |- ( y = x -> <. x , y >. = <. x , x >. )
15	14	eleq1d 2533	. . . . . . . 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 3127	. . . . . . 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 441	. . . . 5 |- ( K e. Preset -> ( x e. B -> E. y <. x , y >. e. ( ( le ` K ) i^i ( B X. B ) ) ) )
19		inss2 3644	. . . . . . . 8 |- ( ( le ` K ) i^i ( B X. B ) ) C_ ( B X. B )
20	19	sseli 3414	. . . . . . 7 |- ( <. x , y >. e. ( ( le ` K ) i^i ( B X. B ) ) -> <. x , y >. e. ( B X. B ) )
21		opelxp1 4872	. . . . . . 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 1784	. . . . 5 |- ( E. y <. x , y >. e. ( ( le ` K ) i^i ( B X. B ) ) -> x e. B )
24	18, 23	impbid1 208	. . . 4 |- ( K e. Preset -> ( x e. B <-> E. y <. x , y >. e. ( ( le ` K ) i^i ( B X. B ) ) ) )
25	13	eldm2 5038	. . . 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 272	. . 3 |- ( K e. Preset -> ( x e. dom ( ( le ` K ) i^i ( B X. B ) ) <-> x e. B ) )
27	3, 26	syl5bb 265	. 2 |- ( K e. Preset -> ( x e. dom .<_ <-> x e. B ) )
28	27	eqrdv 2469	1 |- ( K e. Preset -> dom .<_ = B )

Syntax hints:   -> wi 4   /\ wa 376   = wceq 1452   E.wex 1671   e. wcel 1904   i^i cin 3389   <.cop 3965   class class class wbr 4395   X. cxp 4837   dom cdm 4839   ` cfv 5589   Basecbs 15199   lecple 15275   Preset cpreset 16249

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-xp 4845  df-dm 4849  df-iota 5553  df-fv 5597  df-preset 16251

This theorem is referenced by:  prsssdm  28797  ordtprsval  28798  ordtprsuni  28799  ordtrestNEW  28801  ordtconlem1  28804