Theorem cvrfval 16987

Description: Value of covers relation "is covered by".

Hypotheses

Ref	Expression
cvrfval.b	|- B = (base` K)
cvrfval.s	|- S = (lt` K)
cvrfval.c	|- C = ( <oNEW ` K)

Assertion

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

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

Proof of Theorem cvrfval

Step	Hyp	Ref	Expression
1		elisset 2299	. 2 |- (K e. A -> K e. _V)
2		fveq2 4681	. . . . . . . . 9 |- (p = K -> (base` p) = (base` K))
3		cvrfval.b	. . . . . . . . 9 |- B = (base` K)
4	2, 3	syl6eqr 1946	. . . . . . . 8 |- (p = K -> (base` p) = B)
5	4	eleq2d 1964	. . . . . . 7 |- (p = K -> (x e. (base` p) <-> x e. B))
6	4	eleq2d 1964	. . . . . . 7 |- (p = K -> (y e. (base` p) <-> y e. B))
7	5, 6	anbi12d 690	. . . . . 6 |- (p = K -> ((x e. (base` p) /\ y e. (base` p)) <-> (x e. B /\ y e. B)))
8		fveq2 4681	. . . . . . . 8 |- (p = K -> (lt` p) = (lt` K))
9		cvrfval.s	. . . . . . . 8 |- S = (lt` K)
10	8, 9	syl6eqr 1946	. . . . . . 7 |- (p = K -> (lt` p) = S)
11	10	breqd 3349	. . . . . 6 |- (p = K -> (x(lt` p)y <-> xSy))
12	10	breqd 3349	. . . . . . . . 9 |- (p = K -> (x(lt` p)z <-> xSz))
13	10	breqd 3349	. . . . . . . . 9 |- (p = K -> (z(lt` p)y <-> zSy))
14	12, 13	anbi12d 690	. . . . . . . 8 |- (p = K -> ((x(lt` p)z /\ z(lt` p)y) <-> (xSz /\ zSy)))
15	4, 14	rexeqbidv 2275	. . . . . . 7 |- (p = K -> (E.z e. (base` p)(x(lt` p)z /\ z(lt` p)y) <-> E.z e. B (xSz /\ zSy)))
16	15	notbid 673	. . . . . 6 |- (p = K -> (-. E.z e. (base` p)(x(lt` p)z /\ z(lt` p)y) <-> -. E.z e. B (xSz /\ zSy)))
17	7, 11, 16	3anbi123d 1168	. . . . 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) /\ xSy /\ -. E.z e. B (xSz /\ zSy))))
18	17	opabbidv 3401	. . . 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) /\ xSy /\ -. E.z e. B (xSz /\ zSy))})
19		df-covers 16984	. . . 4 |- <oNEW = (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))})
20		3anass 862	. . . . . 6 |- (((x e. B /\ y e. B) /\ xSy /\ -. E.z e. B (xSz /\ zSy)) <-> ((x e. B /\ y e. B) /\ (xSy /\ -. E.z e. B (xSz /\ zSy))))
21	20	opabbii 3402	. . . . 5 |- {<.x, y>. | ((x e. B /\ y e. B) /\ xSy /\ -. E.z e. B (xSz /\ zSy))} = {<.x, y>. | ((x e. B /\ y e. B) /\ (xSy /\ -. E.z e. B (xSz /\ zSy)))}
22		fvex 4689	. . . . . . . 8 |- (base` K) e. _V
23	3, 22	eqeltri 1967	. . . . . . 7 |- B e. _V
24	23, 23	xpex 4096	. . . . . 6 |- (B X. B) e. _V
25		opabssxp 4060	. . . . . 6 |- {<.x, y>. | ((x e. B /\ y e. B) /\ (xSy /\ -. E.z e. B (xSz /\ zSy)))} C_ (B X. B)
26	24, 25	ssexi 3456	. . . . 5 |- {<.x, y>. | ((x e. B /\ y e. B) /\ (xSy /\ -. E.z e. B (xSz /\ zSy)))} e. _V
27	21, 26	eqeltri 1967	. . . 4 |- {<.x, y>. | ((x e. B /\ y e. B) /\ xSy /\ -. E.z e. B (xSz /\ zSy))} e. _V
28	18, 19, 27	fvmpt 5015	. . 3 |- (K e. _V -> ( <oNEW ` K) = {<.x, y>. | ((x e. B /\ y e. B) /\ xSy /\ -. E.z e. B (xSz /\ zSy))})
29		cvrfval.c	. . 3 |- C = ( <oNEW ` K)
30	28, 29	syl5eq 1940	. 2 |- (K e. _V -> C = {<.x, y>. | ((x e. B /\ y e. B) /\ xSy /\ -. E.z e. B (xSz /\ zSy))})
31	1, 30	syl 12	1 |- (K e. A -> C = {<.x, y>. | ((x e. B /\ y e. B) /\ xSy /\ -. E.z e. B (xSz /\ zSy))})

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wrex 2106  _Vcvv 2292   class class class wbr 3338  {copab 3395   X. cxp 3984  ` cfv 3998  basecbs 16758  ltcplt 16761   <o_NEW ccvr 16980

This theorem is referenced by:  cvrval 16988

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-mpt 5006  df-covers 16984

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