Theorem prsrn 29961

Description: Range of the relation of a preset. (Contributed by Thierry Arnoux, 11-Sep-2018.)

Hypotheses

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

Assertion

Ref	Expression
prsrn	|- ( K e. Preset -> ran .<_ = B )

Proof of Theorem prsrn

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	rneqi 5352	. . . 4 |- ran .<_ = ran ( ( le ` K ) i^i ( B X. B ) )
3	2	eleq2i 2693	. . 3 |- ( x e. ran .<_ <-> x e. ran ( ( le ` K ) i^i ( B X. B ) ) )
4		ordtNEW.b	. . . . . . . . . 10 |- B = ( Base ` K )
5		eqid 2622	. . . . . . . . . 10 |- ( le ` K ) = ( le ` K )
6	4, 5	prsref 16932	. . . . . . . . 9 |- ( ( K e. Preset /\ x e. B ) -> x ( le ` K ) x )
7		df-br 4654	. . . . . . . . 9 |- ( x ( le ` K ) x <-> <. x , x >. e. ( le ` K ) )
8	6, 7	sylib 208	. . . . . . . 8 |- ( ( K e. Preset /\ x e. B ) -> <. x , x >. e. ( le ` K ) )
9		simpr 477	. . . . . . . . 9 |- ( ( K e. Preset /\ x e. B ) -> x e. B )
10		opelxpi 5148	. . . . . . . . 9 |- ( ( x e. B /\ x e. B ) -> <. x , x >. e. ( B X. B ) )
11	9, 10	sylancom 701	. . . . . . . 8 |- ( ( K e. Preset /\ x e. B ) -> <. x , x >. e. ( B X. B ) )
12	8, 11	elind 3798	. . . . . . 7 |- ( ( K e. Preset /\ x e. B ) -> <. x , x >. e. ( ( le ` K ) i^i ( B X. B ) ) )
13		vex 3203	. . . . . . . 8 |- x e. _V
14		opeq1 4402	. . . . . . . . 9 |- ( y = x -> <. y , x >. = <. x , x >. )
15	14	eleq1d 2686	. . . . . . . 8 |- ( y = x -> ( <. y , x >. e. ( ( le ` K ) i^i ( B X. B ) ) <-> <. x , x >. e. ( ( le ` K ) i^i ( B X. B ) ) ) )
16	13, 15	spcev 3300	. . . . . . 7 |- ( <. x , x >. e. ( ( le ` K ) i^i ( B X. B ) ) -> E. y <. y , x >. e. ( ( le ` K ) i^i ( B X. B ) ) )
17	12, 16	syl 17	. . . . . 6 |- ( ( K e. Preset /\ x e. B ) -> E. y <. y , x >. e. ( ( le ` K ) i^i ( B X. B ) ) )
18	17	ex 450	. . . . 5 |- ( K e. Preset -> ( x e. B -> E. y <. y , x >. e. ( ( le ` K ) i^i ( B X. B ) ) ) )
19		inss2 3834	. . . . . . . 8 |- ( ( le ` K ) i^i ( B X. B ) ) C_ ( B X. B )
20	19	sseli 3599	. . . . . . 7 |- ( <. y , x >. e. ( ( le ` K ) i^i ( B X. B ) ) -> <. y , x >. e. ( B X. B ) )
21		opelxp2 5151	. . . . . . 7 |- ( <. y , x >. e. ( B X. B ) -> x e. B )
22	20, 21	syl 17	. . . . . 6 |- ( <. y , x >. e. ( ( le ` K ) i^i ( B X. B ) ) -> x e. B )
23	22	exlimiv 1858	. . . . 5 |- ( E. y <. y , x >. e. ( ( le ` K ) i^i ( B X. B ) ) -> x e. B )
24	18, 23	impbid1 215	. . . 4 |- ( K e. Preset -> ( x e. B <-> E. y <. y , x >. e. ( ( le ` K ) i^i ( B X. B ) ) ) )
25	13	elrn2 5365	. . . 4 |- ( x e. ran ( ( le ` K ) i^i ( B X. B ) ) <-> E. y <. y , x >. e. ( ( le ` K ) i^i ( B X. B ) ) )
26	24, 25	syl6rbbr 279	. . 3 |- ( K e. Preset -> ( x e. ran ( ( le ` K ) i^i ( B X. B ) ) <-> x e. B ) )
27	3, 26	syl5bb 272	. 2 |- ( K e. Preset -> ( x e. ran .<_ <-> x e. B ) )
28	27	eqrdv 2620	1 |- ( K e. Preset -> ran .<_ = B )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   i^i cin 3573   <.cop 4183   class class class wbr 4653   X. cxp 5112   ran crn 5115   ` cfv 5888   Basecbs 15857   lecple 15948   Preset cpreset 16926

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-sbc 3436  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-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-iota 5851  df-fv 5896  df-preset 16928

This theorem is referenced by: (None)