Theorem cvrfval 34555

Description: Value of covers relation "is covered by". (Contributed by NM, 18-Sep-2011.)

Hypotheses

Ref	Expression
cvrfval.b	|- B = ( Base ` K )
cvrfval.s	|- .< = ( lt ` K )
cvrfval.c	|- C = ( <o ` K )

Assertion

Ref	Expression
cvrfval	|- ( K e. A -> C = { <. x , y >. | ( ( x e. B /\ y e. B ) /\ x .< y /\ -. E. z e. B ( x .< z /\ z .< y ) ) } )

Distinct variable groups:   x, y, z, B   x, K, y, z

Allowed substitution hints:   A( x, y, z)   C( x, y, z)   .< ( x, y, z)

Proof of Theorem cvrfval

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( K e. A -> K e. _V )
2		cvrfval.c	. . 3 |- C = ( <o ` K )
3		fveq2 6191	. . . . . . . . 9 |- ( p = K -> ( Base ` p ) = ( Base ` K ) )
4		cvrfval.b	. . . . . . . . 9 |- B = ( Base ` K )
5	3, 4	syl6eqr 2674	. . . . . . . 8 |- ( p = K -> ( Base ` p ) = B )
6	5	eleq2d 2687	. . . . . . 7 |- ( p = K -> ( x e. ( Base ` p ) <-> x e. B ) )
7	5	eleq2d 2687	. . . . . . 7 |- ( p = K -> ( y e. ( Base ` p ) <-> y e. B ) )
8	6, 7	anbi12d 747	. . . . . 6 |- ( p = K -> ( ( x e. ( Base ` p ) /\ y e. ( Base ` p ) ) <-> ( x e. B /\ y e. B ) ) )
9		fveq2 6191	. . . . . . . 8 |- ( p = K -> ( lt ` p ) = ( lt ` K ) )
10		cvrfval.s	. . . . . . . 8 |- .< = ( lt ` K )
11	9, 10	syl6eqr 2674	. . . . . . 7 |- ( p = K -> ( lt ` p ) = .< )
12	11	breqd 4664	. . . . . 6 |- ( p = K -> ( x ( lt ` p ) y <-> x .< y ) )
13	11	breqd 4664	. . . . . . . . 9 |- ( p = K -> ( x ( lt ` p ) z <-> x .< z ) )
14	11	breqd 4664	. . . . . . . . 9 |- ( p = K -> ( z ( lt ` p ) y <-> z .< y ) )
15	13, 14	anbi12d 747	. . . . . . . 8 |- ( p = K -> ( ( x ( lt ` p ) z /\ z ( lt ` p ) y ) <-> ( x .< z /\ z .< y ) ) )
16	5, 15	rexeqbidv 3153	. . . . . . 7 |- ( p = K -> ( E. z e. ( Base ` p ) ( x ( lt ` p ) z /\ z ( lt ` p ) y ) <-> E. z e. B ( x .< z /\ z .< y ) ) )
17	16	notbid 308	. . . . . 6 |- ( p = K -> ( -. E. z e. ( Base ` p ) ( x ( lt ` p ) z /\ z ( lt ` p ) y ) <-> -. E. z e. B ( x .< z /\ z .< y ) ) )
18	8, 12, 17	3anbi123d 1399	. . . . 5 |- ( p = K -> ( ( ( x e. ( Base ` p ) /\ y e. ( Base ` p ) ) /\ x ( lt ` p ) y /\ -. E. z e. ( Base ` p ) ( x ( lt ` p ) z /\ z ( lt ` p ) y ) ) <-> ( ( x e. B /\ y e. B ) /\ x .< y /\ -. E. z e. B ( x .< z /\ z .< y ) ) ) )
19	18	opabbidv 4716	. . . 4 |- ( p = K -> { <. x , y >. | ( ( x e. ( Base ` p ) /\ y e. ( Base ` p ) ) /\ x ( lt ` p ) y /\ -. E. z e. ( Base ` p ) ( x ( lt ` p ) z /\ z ( lt ` p ) y ) ) } = { <. x , y >. | ( ( x e. B /\ y e. B ) /\ x .< y /\ -. E. z e. B ( x .< z /\ z .< y ) ) } )
20		df-covers 34553	. . . 4 |- <o = ( p e. _V |-> { <. x , y >. | ( ( x e. ( Base ` p ) /\ y e. ( Base ` p ) ) /\ x ( lt ` p ) y /\ -. E. z e. ( Base ` p ) ( x ( lt ` p ) z /\ z ( lt ` p ) y ) ) } )
21		3anass 1042	. . . . . 6 |- ( ( ( x e. B /\ y e. B ) /\ x .< y /\ -. E. z e. B ( x .< z /\ z .< y ) ) <-> ( ( x e. B /\ y e. B ) /\ ( x .< y /\ -. E. z e. B ( x .< z /\ z .< y ) ) ) )
22	21	opabbii 4717	. . . . 5 |- { <. x , y >. | ( ( x e. B /\ y e. B ) /\ x .< y /\ -. E. z e. B ( x .< z /\ z .< y ) ) } = { <. x , y >. | ( ( x e. B /\ y e. B ) /\ ( x .< y /\ -. E. z e. B ( x .< z /\ z .< y ) ) ) }
23		fvex 6201	. . . . . . . 8 |- ( Base ` K ) e. _V
24	4, 23	eqeltri 2697	. . . . . . 7 |- B e. _V
25	24, 24	xpex 6962	. . . . . 6 |- ( B X. B ) e. _V
26		opabssxp 5193	. . . . . 6 |- { <. x , y >. | ( ( x e. B /\ y e. B ) /\ ( x .< y /\ -. E. z e. B ( x .< z /\ z .< y ) ) ) } C_ ( B X. B )
27	25, 26	ssexi 4803	. . . . 5 |- { <. x , y >. | ( ( x e. B /\ y e. B ) /\ ( x .< y /\ -. E. z e. B ( x .< z /\ z .< y ) ) ) } e. _V
28	22, 27	eqeltri 2697	. . . 4 |- { <. x , y >. | ( ( x e. B /\ y e. B ) /\ x .< y /\ -. E. z e. B ( x .< z /\ z .< y ) ) } e. _V
29	19, 20, 28	fvmpt 6282	. . 3 |- ( K e. _V -> ( <o ` K ) = { <. x , y >. | ( ( x e. B /\ y e. B ) /\ x .< y /\ -. E. z e. B ( x .< z /\ z .< y ) ) } )
30	2, 29	syl5eq 2668	. 2 |- ( K e. _V -> C = { <. x , y >. | ( ( x e. B /\ y e. B ) /\ x .< y /\ -. E. z e. B ( x .< z /\ z .< y ) ) } )
31	1, 30	syl 17	1 |- ( K e. A -> C = { <. x , y >. | ( ( x e. B /\ y e. B ) /\ x .< y /\ -. E. z e. B ( x .< z /\ z .< y ) ) } )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   class class class wbr 4653   {copab 4712   X. cxp 5112   ` cfv 5888   Basecbs 15857   ltcplt 16941   <o ccvr 34549

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-covers 34553

This theorem is referenced by:  cvrval  34556