Theorem isclat 17109

Description: The predicate "is a complete lattice." (Contributed by NM, 18-Oct-2012.) (Revised by NM, 12-Sep-2018.)

Hypotheses

Ref	Expression
isclat.b	|- B = ( Base ` K )
isclat.u	|- U = ( lub ` K )
isclat.g	|- G = ( glb ` K )

Assertion

Ref	Expression
isclat	|- ( K e. CLat <-> ( K e. Poset /\ ( dom U = ~P B /\ dom G = ~P B ) ) )

Proof of Theorem isclat

Dummy variable l is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . 6 |- ( l = K -> ( lub ` l ) = ( lub ` K ) )
2		isclat.u	. . . . . 6 |- U = ( lub ` K )
3	1, 2	syl6eqr 2674	. . . . 5 |- ( l = K -> ( lub ` l ) = U )
4	3	dmeqd 5326	. . . 4 |- ( l = K -> dom ( lub ` l ) = dom U )
5		fveq2 6191	. . . . . 6 |- ( l = K -> ( Base ` l ) = ( Base ` K ) )
6		isclat.b	. . . . . 6 |- B = ( Base ` K )
7	5, 6	syl6eqr 2674	. . . . 5 |- ( l = K -> ( Base ` l ) = B )
8	7	pweqd 4163	. . . 4 |- ( l = K -> ~P ( Base ` l ) = ~P B )
9	4, 8	eqeq12d 2637	. . 3 |- ( l = K -> ( dom ( lub ` l ) = ~P ( Base ` l ) <-> dom U = ~P B ) )
10		fveq2 6191	. . . . . 6 |- ( l = K -> ( glb ` l ) = ( glb ` K ) )
11		isclat.g	. . . . . 6 |- G = ( glb ` K )
12	10, 11	syl6eqr 2674	. . . . 5 |- ( l = K -> ( glb ` l ) = G )
13	12	dmeqd 5326	. . . 4 |- ( l = K -> dom ( glb ` l ) = dom G )
14	13, 8	eqeq12d 2637	. . 3 |- ( l = K -> ( dom ( glb ` l ) = ~P ( Base ` l ) <-> dom G = ~P B ) )
15	9, 14	anbi12d 747	. 2 |- ( l = K -> ( ( dom ( lub ` l ) = ~P ( Base ` l ) /\ dom ( glb ` l ) = ~P ( Base ` l ) ) <-> ( dom U = ~P B /\ dom G = ~P B ) ) )
16		df-clat 17108	. 2 |- CLat = { l e. Poset | ( dom ( lub ` l ) = ~P ( Base ` l ) /\ dom ( glb ` l ) = ~P ( Base ` l ) ) }
17	15, 16	elrab2 3366	1 |- ( K e. CLat <-> ( K e. Poset /\ ( dom U = ~P B /\ dom G = ~P B ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   ~Pcpw 4158   dom cdm 5114   ` cfv 5888   Basecbs 15857   Posetcpo 16940   lubclub 16942   glbcglb 16943   CLatccla 17107

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  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-dm 5124  df-iota 5851  df-fv 5896  df-clat 17108

This theorem is referenced by:  clatpos  17110  clatlem  17111  clatlubcl2  17113  clatglbcl2  17115  clatl  17116  oduclatb  17144  xrsclat  29680