Theorem lubfval 16803

Description: Value of least upper bound function of a poset.

Hypotheses

Ref	Expression
lubfval.b	|- B = (base` K)
lubfval.l	|- L = (le` K)
lubfval.u	|- U = (lub` K)

Assertion

Ref	Expression
lubfval	|- (K e. A -> U = (s e. ~PB |-> (iota_x e. B(A.y e. s yLx /\ A.z e. B (A.y e. s yLz -> xLz)))))

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

Proof of Theorem lubfval

Step	Hyp	Ref	Expression
1		elisset 2299	. 2 |- (K e. A -> K e. _V)
2		fveq2 4681	. . . . . . 7 |- (p = K -> (base` p) = (base` K))
3		lubfval.b	. . . . . . 7 |- B = (base` K)
4	2, 3	syl6eqr 1946	. . . . . 6 |- (p = K -> (base` p) = B)
5		pweq 3036	. . . . . 6 |- ((base` p) = B -> ~P(base` p) = ~PB)
6	4, 5	syl 12	. . . . 5 |- (p = K -> ~P(base` p) = ~PB)
7		fveq2 4681	. . . . . . . . . 10 |- (p = K -> (le` p) = (le` K))
8		lubfval.l	. . . . . . . . . 10 |- L = (le` K)
9	7, 8	syl6eqr 1946	. . . . . . . . 9 |- (p = K -> (le` p) = L)
10	9	breqd 3349	. . . . . . . 8 |- (p = K -> (y(le` p)x <-> yLx))
11	10	ralbidv 2123	. . . . . . 7 |- (p = K -> (A.y e. s y(le` p)x <-> A.y e. s yLx))
12	9	breqd 3349	. . . . . . . . . 10 |- (p = K -> (y(le` p)z <-> yLz))
13	12	ralbidv 2123	. . . . . . . . 9 |- (p = K -> (A.y e. s y(le` p)z <-> A.y e. s yLz))
14	9	breqd 3349	. . . . . . . . 9 |- (p = K -> (x(le` p)z <-> xLz))
15	13, 14	imbi12d 688	. . . . . . . 8 |- (p = K -> ((A.y e. s y(le` p)z -> x(le` p)z) <-> (A.y e. s yLz -> xLz)))
16	4, 15	raleqbidv 2274	. . . . . . 7 |- (p = K -> (A.z e. (base` p)(A.y e. s y(le` p)z -> x(le` p)z) <-> A.z e. B (A.y e. s yLz -> xLz)))
17	11, 16	anbi12d 690	. . . . . 6 |- (p = K -> ((A.y e. s y(le` p)x /\ A.z e. (base` p)(A.y e. s y(le` p)z -> x(le` p)z)) <-> (A.y e. s yLx /\ A.z e. B (A.y e. s yLz -> xLz))))
18	4, 17	riotaeqbidv 5563	. . . . 5 |- (p = K -> (iota_x e. (base` p)(A.y e. s y(le` p)x /\ A.z e. (base` p)(A.y e. s y(le` p)z -> x(le` p)z))) = (iota_x e. B(A.y e. s yLx /\ A.z e. B (A.y e. s yLz -> xLz))))
19	6, 18	mpteq12dv 5008	. . . 4 |- (p = K -> (s e. ~P(base` p) |-> (iota_x e. (base` p)(A.y e. s y(le` p)x /\ A.z e. (base` p)(A.y e. s y(le` p)z -> x(le` p)z)))) = (s e. ~PB |-> (iota_x e. B(A.y e. s yLx /\ A.z e. B (A.y e. s yLz -> xLz)))))
20		df-lub 16799	. . . 4 |- lub = (p e. _V |-> (s e. ~P(base` p) |-> (iota_x e. (base` p)(A.y e. s y(le` p)x /\ A.z e. (base` p)(A.y e. s y(le` p)z -> x(le` p)z)))))
21		fvex 4689	. . . . . . 7 |- (base` K) e. _V
22	3, 21	eqeltri 1967	. . . . . 6 |- B e. _V
23	22	pwex 3487	. . . . 5 |- ~PB e. _V
24		mptexg 5012	. . . . 5 |- (~PB e. _V -> (s e. ~PB |-> (iota_x e. B(A.y e. s yLx /\ A.z e. B (A.y e. s yLz -> xLz)))) e. _V)
25	23, 24	ax-mp 7	. . . 4 |- (s e. ~PB |-> (iota_x e. B(A.y e. s yLx /\ A.z e. B (A.y e. s yLz -> xLz)))) e. _V
26	19, 20, 25	fvmpt 5015	. . 3 |- (K e. _V -> (lub` K) = (s e. ~PB |-> (iota_x e. B(A.y e. s yLx /\ A.z e. B (A.y e. s yLz -> xLz)))))
27		lubfval.u	. . 3 |- U = (lub` K)
28	26, 27	syl5eq 1940	. 2 |- (K e. _V -> U = (s e. ~PB |-> (iota_x e. B(A.y e. s yLx /\ A.z e. B (A.y e. s yLz -> xLz)))))
29	1, 28	syl 12	1 |- (K e. A -> U = (s e. ~PB |-> (iota_x e. B(A.y e. s yLx /\ A.z e. B (A.y e. s yLz -> xLz)))))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292  ~Pcpw 3032   class class class wbr 3338  ` cfv 3998   e. cmpt 5004  iota_crio 5555  basecbs 16758  lecple 16759  lubclub 16764

This theorem is referenced by:  lubval 16804

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-rep 3428  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-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  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-fn 4009  df-fv 4014  df-mpt 5006  df-iota 5089  df-riota 5560  df-lub 16799

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