Theorem joinfval 16814

Description: Value of join function for a poset.

Hypotheses

Ref	Expression
joinfval.b	|- B = (base` K)
joinfval.u	|- U = (lub` K)
joinfval.j	|- J = (join` K)

Assertion

Ref	Expression
joinfval	|- (K e. A -> J = (x e. B, y e. B |-> (U` {x, y})))

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

Proof of Theorem joinfval

Step	Hyp	Ref	Expression
1		elisset 2299	. 2 |- (K e. A -> K e. _V)
2		fveq2 4681	. . . . . 6 |- (p = K -> (base` p) = (base` K))
3		joinfval.b	. . . . . 6 |- B = (base` K)
4	2, 3	syl6eqr 1946	. . . . 5 |- (p = K -> (base` p) = B)
5		fveq2 4681	. . . . . . 7 |- (p = K -> (lub` p) = (lub` K))
6		joinfval.u	. . . . . . 7 |- U = (lub` K)
7	5, 6	syl6eqr 1946	. . . . . 6 |- (p = K -> (lub` p) = U)
8	7	fveq1d 4683	. . . . 5 |- (p = K -> ((lub` p)` {x, y}) = (U` {x, y}))
9	4, 4, 8	mpt2eq123dv 5009	. . . 4 |- (p = K -> (x e. (base` p), y e. (base` p) |-> ((lub` p)` {x, y})) = (x e. B, y e. B |-> (U` {x, y})))
10		df-join 16801	. . . 4 |- join = (p e. _V |-> (x e. (base` p), y e. (base` p) |-> ((lub` p)` {x, y})))
11		fvex 4689	. . . . . 6 |- (base` K) e. _V
12	3, 11	eqeltri 1967	. . . . 5 |- B e. _V
13		mpt2exg 5013	. . . . 5 |- ((B e. _V /\ B e. _V) -> (x e. B, y e. B |-> (U` {x, y})) e. _V)
14	12, 12, 13	mp2an 761	. . . 4 |- (x e. B, y e. B |-> (U` {x, y})) e. _V
15	9, 10, 14	fvmpt 5015	. . 3 |- (K e. _V -> (join` K) = (x e. B, y e. B |-> (U` {x, y})))
16		joinfval.j	. . 3 |- J = (join` K)
17	15, 16	syl5eq 1940	. 2 |- (K e. _V -> J = (x e. B, y e. B |-> (U` {x, y})))
18	1, 17	syl 12	1 |- (K e. A -> J = (x e. B, y e. B |-> (U` {x, y})))

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300  _Vcvv 2292  {cpr 3045  ` cfv 3998   e. cmpt2 5005  basecbs 16758  lubclub 16764  joincjn 16766

This theorem is referenced by:  joinval 16815

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-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-fn 4009  df-fv 4014  df-oprab 4887  df-mpt 5006  df-mpt2 5007  df-join 16801

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