Theorem meetfval 17015

Description: Value of meet function for a poset. (Contributed by NM, 12-Sep-2011.) (Revised by NM, 9-Sep-2018.) TODO: prove meetfval2 17016 first to reduce net proof size (existence part)?

Hypotheses

Ref	Expression
meetfval.u	|- G = ( glb ` K )
meetfval.m	|- ./\ = ( meet ` K )

Assertion

Ref	Expression
meetfval	|- ( K e. V -> ./\ = { <. <. x , y >. , z >. | { x , y } G z } )

Distinct variable groups:   x, y, z, K   z, G

Allowed substitution hints:   G( x, y)   ./\ ( x, y, z)   V( x, y, z)

Proof of Theorem meetfval

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( K e. V -> K e. _V )
2		meetfval.m	. . 3 |- ./\ = ( meet ` K )
3		fvex 6201	. . . . . . 7 |- ( Base ` K ) e. _V
4		moeq 3382	. . . . . . . 8 |- E* z z = ( G ` { x , y } )
5	4	a1i 11	. . . . . . 7 |- ( ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) -> E* z z = ( G ` { x , y } ) )
6		eqid 2622	. . . . . . 7 |- { <. <. x , y >. , z >. | ( ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) /\ z = ( G ` { x , y } ) ) } = { <. <. x , y >. , z >. | ( ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) /\ z = ( G ` { x , y } ) ) }
7	3, 3, 5, 6	oprabex 7156	. . . . . 6 |- { <. <. x , y >. , z >. | ( ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) /\ z = ( G ` { x , y } ) ) } e. _V
8	7	a1i 11	. . . . 5 |- ( K e. _V -> { <. <. x , y >. , z >. | ( ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) /\ z = ( G ` { x , y } ) ) } e. _V )
9		meetfval.u	. . . . . . . . . . . 12 |- G = ( glb ` K )
10	9	glbfun 16993	. . . . . . . . . . 11 |- Fun G
11		funbrfv2b 6240	. . . . . . . . . . 11 |- ( Fun G -> ( { x , y } G z <-> ( { x , y } e. dom G /\ ( G ` { x , y } ) = z ) ) )
12	10, 11	ax-mp 5	. . . . . . . . . 10 |- ( { x , y } G z <-> ( { x , y } e. dom G /\ ( G ` { x , y } ) = z ) )
13		eqid 2622	. . . . . . . . . . . . . 14 |- ( Base ` K ) = ( Base ` K )
14		eqid 2622	. . . . . . . . . . . . . 14 |- ( le ` K ) = ( le ` K )
15		simpl 473	. . . . . . . . . . . . . 14 |- ( ( K e. _V /\ { x , y } e. dom G ) -> K e. _V )
16		simpr 477	. . . . . . . . . . . . . 14 |- ( ( K e. _V /\ { x , y } e. dom G ) -> { x , y } e. dom G )
17	13, 14, 9, 15, 16	glbelss 16995	. . . . . . . . . . . . 13 |- ( ( K e. _V /\ { x , y } e. dom G ) -> { x , y } C_ ( Base ` K ) )
18	17	ex 450	. . . . . . . . . . . 12 |- ( K e. _V -> ( { x , y } e. dom G -> { x , y } C_ ( Base ` K ) ) )
19		vex 3203	. . . . . . . . . . . . 13 |- x e. _V
20		vex 3203	. . . . . . . . . . . . 13 |- y e. _V
21	19, 20	prss 4351	. . . . . . . . . . . 12 |- ( ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) <-> { x , y } C_ ( Base ` K ) )
22	18, 21	syl6ibr 242	. . . . . . . . . . 11 |- ( K e. _V -> ( { x , y } e. dom G -> ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) ) )
23		eqcom 2629	. . . . . . . . . . . . 13 |- ( ( G ` { x , y } ) = z <-> z = ( G ` { x , y } ) )
24	23	biimpi 206	. . . . . . . . . . . 12 |- ( ( G ` { x , y } ) = z -> z = ( G ` { x , y } ) )
25	24	a1i 11	. . . . . . . . . . 11 |- ( K e. _V -> ( ( G ` { x , y } ) = z -> z = ( G ` { x , y } ) ) )
26	22, 25	anim12d 586	. . . . . . . . . 10 |- ( K e. _V -> ( ( { x , y } e. dom G /\ ( G ` { x , y } ) = z ) -> ( ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) /\ z = ( G ` { x , y } ) ) ) )
27	12, 26	syl5bi 232	. . . . . . . . 9 |- ( K e. _V -> ( { x , y } G z -> ( ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) /\ z = ( G ` { x , y } ) ) ) )
28	27	alrimiv 1855	. . . . . . . 8 |- ( K e. _V -> A. z ( { x , y } G z -> ( ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) /\ z = ( G ` { x , y } ) ) ) )
29	28	alrimiv 1855	. . . . . . 7 |- ( K e. _V -> A. y A. z ( { x , y } G z -> ( ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) /\ z = ( G ` { x , y } ) ) ) )
30	29	alrimiv 1855	. . . . . 6 |- ( K e. _V -> A. x A. y A. z ( { x , y } G z -> ( ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) /\ z = ( G ` { x , y } ) ) ) )
31		ssoprab2 6711	. . . . . 6 |- ( A. x A. y A. z ( { x , y } G z -> ( ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) /\ z = ( G ` { x , y } ) ) ) -> { <. <. x , y >. , z >. | { x , y } G z } C_ { <. <. x , y >. , z >. | ( ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) /\ z = ( G ` { x , y } ) ) } )
32	30, 31	syl 17	. . . . 5 |- ( K e. _V -> { <. <. x , y >. , z >. | { x , y } G z } C_ { <. <. x , y >. , z >. | ( ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) /\ z = ( G ` { x , y } ) ) } )
33	8, 32	ssexd 4805	. . . 4 |- ( K e. _V -> { <. <. x , y >. , z >. | { x , y } G z } e. _V )
34		fveq2 6191	. . . . . . . 8 |- ( p = K -> ( glb ` p ) = ( glb ` K ) )
35	34, 9	syl6eqr 2674	. . . . . . 7 |- ( p = K -> ( glb ` p ) = G )
36	35	breqd 4664	. . . . . 6 |- ( p = K -> ( { x , y } ( glb ` p ) z <-> { x , y } G z ) )
37	36	oprabbidv 6709	. . . . 5 |- ( p = K -> { <. <. x , y >. , z >. | { x , y } ( glb ` p ) z } = { <. <. x , y >. , z >. | { x , y } G z } )
38		df-meet 16977	. . . . 5 |- meet = ( p e. _V |-> { <. <. x , y >. , z >. | { x , y } ( glb ` p ) z } )
39	37, 38	fvmptg 6280	. . . 4 |- ( ( K e. _V /\ { <. <. x , y >. , z >. | { x , y } G z } e. _V ) -> ( meet ` K ) = { <. <. x , y >. , z >. | { x , y } G z } )
40	33, 39	mpdan 702	. . 3 |- ( K e. _V -> ( meet ` K ) = { <. <. x , y >. , z >. | { x , y } G z } )
41	2, 40	syl5eq 2668	. 2 |- ( K e. _V -> ./\ = { <. <. x , y >. , z >. | { x , y } G z } )
42	1, 41	syl 17	1 |- ( K e. V -> ./\ = { <. <. x , y >. , z >. | { x , y } G z } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   E*wmo 2471   _Vcvv 3200   C_ wss 3574   {cpr 4179   class class class wbr 4653   dom cdm 5114   Fun wfun 5882   ` cfv 5888   {coprab 6651   Basecbs 15857   lecple 15948   glbcglb 16943   meetcmee 16945

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-8 1992  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  ax-un 6949

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-oprab 6654  df-glb 16975  df-meet 16977

This theorem is referenced by:  meetfval2  17016  meet0  17137  odumeet  17140  odujoin  17142