Theorem glbdm 16992

Description: Domain of the greatest lower bound function of a poset. (Contributed by NM, 6-Sep-2018.)

Hypotheses

Ref	Expression
glbfval.b	|- B = ( Base ` K )
glbfval.l	|- .<_ = ( le ` K )
glbfval.g	|- G = ( glb ` K )
glbfval.p	|- ( ps <-> ( A. y e. s x .<_ y /\ A. z e. B ( A. y e. s z .<_ y -> z .<_ x ) ) )
glbfval.k	|- ( ph -> K e. V )

Assertion

Ref	Expression
glbdm	|- ( ph -> dom G = { s e. ~P B | E! x e. B ps } )

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

Allowed substitution hints:   ph( x, y, z, s)   ps( x, y, z, s)   B( y)   G( x, y, z, s)   .<_ ( x, y, z, s)   V( x, y, z, s)

Proof of Theorem glbdm

Step	Hyp	Ref	Expression
1		glbfval.b	. . . 4 |- B = ( Base ` K )
2		glbfval.l	. . . 4 |- .<_ = ( le ` K )
3		glbfval.g	. . . 4 |- G = ( glb ` K )
4		glbfval.p	. . . 4 |- ( ps <-> ( A. y e. s x .<_ y /\ A. z e. B ( A. y e. s z .<_ y -> z .<_ x ) ) )
5		glbfval.k	. . . 4 |- ( ph -> K e. V )
6	1, 2, 3, 4, 5	glbfval 16991	. . 3 |- ( ph -> G = ( ( s e. ~P B |-> ( iota_ x e. B ps ) ) |` { s | E! x e. B ps } ) )
7	6	dmeqd 5326	. 2 |- ( ph -> dom G = dom ( ( s e. ~P B |-> ( iota_ x e. B ps ) ) |` { s | E! x e. B ps } ) )
8		riotaex 6615	. . . . 5 |- ( iota_ x e. B ps ) e. _V
9		eqid 2622	. . . . 5 |- ( s e. ~P B |-> ( iota_ x e. B ps ) ) = ( s e. ~P B |-> ( iota_ x e. B ps ) )
10	8, 9	dmmpti 6023	. . . 4 |- dom ( s e. ~P B |-> ( iota_ x e. B ps ) ) = ~P B
11	10	ineq2i 3811	. . 3 |- ( { s | E! x e. B ps } i^i dom ( s e. ~P B |-> ( iota_ x e. B ps ) ) ) = ( { s | E! x e. B ps } i^i ~P B )
12		dmres 5419	. . 3 |- dom ( ( s e. ~P B |-> ( iota_ x e. B ps ) ) |` { s | E! x e. B ps } ) = ( { s | E! x e. B ps } i^i dom ( s e. ~P B |-> ( iota_ x e. B ps ) ) )
13		dfrab2 3903	. . 3 |- { s e. ~P B | E! x e. B ps } = ( { s | E! x e. B ps } i^i ~P B )
14	11, 12, 13	3eqtr4i 2654	. 2 |- dom ( ( s e. ~P B |-> ( iota_ x e. B ps ) ) |` { s | E! x e. B ps } ) = { s e. ~P B | E! x e. B ps }
15	7, 14	syl6eq 2672	1 |- ( ph -> dom G = { s e. ~P B | E! x e. B ps } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E!wreu 2914   {crab 2916   i^i cin 3573   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   |` cres 5116   ` cfv 5888   iota_crio 6610   Basecbs 15857   lecple 15948   glbcglb 16943

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  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-glb 16975

This theorem is referenced by:  glbeldm  16994  xrsclat  29680