Theorem p1val 16844

Description: Value of poset zero.

Hypotheses

Ref	Expression
p1val.b	|- B = (base` K)
p1val.u	|- U = (lub` K)
p1val.t	|- T = (1.` K)

Assertion

Ref	Expression
p1val	|- (K e. A -> T = (U` B))

Proof of Theorem p1val

Step	Hyp	Ref	Expression
1		elisset 2299	. 2 |- (K e. A -> K e. _V)
2		fveq2 4681	. . . . . 6 |- (k = K -> (lub` k) = (lub` K))
3		p1val.u	. . . . . 6 |- U = (lub` K)
4	2, 3	syl6eqr 1946	. . . . 5 |- (k = K -> (lub` k) = U)
5		fveq2 4681	. . . . . 6 |- (k = K -> (base` k) = (base` K))
6		p1val.b	. . . . . 6 |- B = (base` K)
7	5, 6	syl6eqr 1946	. . . . 5 |- (k = K -> (base` k) = B)
8	4, 7	fveq12d 10152	. . . 4 |- (k = K -> ((lub` k)` (base` k)) = (U` B))
9		df-p1 16842	. . . 4 |- 1. = (k e. _V |-> ((lub` k)` (base` k)))
10		fvex 4689	. . . 4 |- (U` B) e. _V
11	8, 9, 10	fvmpt 5015	. . 3 |- (K e. _V -> (1.` K) = (U` B))
12		p1val.t	. . 3 |- T = (1.` K)
13	11, 12	syl5eq 1940	. 2 |- (K e. _V -> T = (U` B))
14	1, 13	syl 12	1 |- (K e. A -> T = (U` B))

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300  _Vcvv 2292  ` cfv 3998  basecbs 16758  lubclub 16764  1.cp1 16833

This theorem is referenced by:  ple1 16846

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-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-p1 16842

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