Theorem meeteu 17024

Description: Uniqueness of meet of elements in the domain. (Contributed by NM, 12-Sep-2018.)

Hypotheses

Ref	Expression
meetval2.b	|- B = ( Base ` K )
meetval2.l	|- .<_ = ( le ` K )
meetval2.m	|- ./\ = ( meet ` K )
meetval2.k	|- ( ph -> K e. V )
meetval2.x	|- ( ph -> X e. B )
meetval2.y	|- ( ph -> Y e. B )
meetlem.e	|- ( ph -> <. X , Y >. e. dom ./\ )

Assertion

Ref	Expression
meeteu	|- ( ph -> E! x e. B ( ( x .<_ X /\ x .<_ Y ) /\ A. z e. B ( ( z .<_ X /\ z .<_ Y ) -> z .<_ x ) ) )

Distinct variable groups:   x, z, B   x, ./\ , z   x, K, z   x, X, z   x, Y, z   ph, x

Allowed substitution hints:   ph( z)   .<_ ( x, z)   V( x, z)

Proof of Theorem meeteu

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		meetlem.e	. 2 |- ( ph -> <. X , Y >. e. dom ./\ )
2		eqid 2622	. . . 4 |- ( glb ` K ) = ( glb ` K )
3		meetval2.m	. . . 4 |- ./\ = ( meet ` K )
4		meetval2.k	. . . 4 |- ( ph -> K e. V )
5		meetval2.x	. . . 4 |- ( ph -> X e. B )
6		meetval2.y	. . . 4 |- ( ph -> Y e. B )
7	2, 3, 4, 5, 6	meetdef 17018	. . 3 |- ( ph -> ( <. X , Y >. e. dom ./\ <-> { X , Y } e. dom ( glb ` K ) ) )
8		meetval2.b	. . . . . 6 |- B = ( Base ` K )
9		meetval2.l	. . . . . 6 |- .<_ = ( le ` K )
10		biid 251	. . . . . 6 |- ( ( A. y e. { X , Y } x .<_ y /\ A. z e. B ( A. y e. { X , Y } z .<_ y -> z .<_ x ) ) <-> ( A. y e. { X , Y } x .<_ y /\ A. z e. B ( A. y e. { X , Y } z .<_ y -> z .<_ x ) ) )
11	4	adantr 481	. . . . . 6 |- ( ( ph /\ { X , Y } e. dom ( glb ` K ) ) -> K e. V )
12		simpr 477	. . . . . 6 |- ( ( ph /\ { X , Y } e. dom ( glb ` K ) ) -> { X , Y } e. dom ( glb ` K ) )
13	8, 9, 2, 10, 11, 12	glbeu 16996	. . . . 5 |- ( ( ph /\ { X , Y } e. dom ( glb ` K ) ) -> E! x e. B ( A. y e. { X , Y } x .<_ y /\ A. z e. B ( A. y e. { X , Y } z .<_ y -> z .<_ x ) ) )
14	13	ex 450	. . . 4 |- ( ph -> ( { X , Y } e. dom ( glb ` K ) -> E! x e. B ( A. y e. { X , Y } x .<_ y /\ A. z e. B ( A. y e. { X , Y } z .<_ y -> z .<_ x ) ) ) )
15	8, 9, 3, 4, 5, 6	meetval2lem 17022	. . . . . 6 |- ( ( X e. B /\ Y e. B ) -> ( ( A. y e. { X , Y } x .<_ y /\ A. z e. B ( A. y e. { X , Y } z .<_ y -> z .<_ x ) ) <-> ( ( x .<_ X /\ x .<_ Y ) /\ A. z e. B ( ( z .<_ X /\ z .<_ Y ) -> z .<_ x ) ) ) )
16	5, 6, 15	syl2anc 693	. . . . 5 |- ( ph -> ( ( A. y e. { X , Y } x .<_ y /\ A. z e. B ( A. y e. { X , Y } z .<_ y -> z .<_ x ) ) <-> ( ( x .<_ X /\ x .<_ Y ) /\ A. z e. B ( ( z .<_ X /\ z .<_ Y ) -> z .<_ x ) ) ) )
17	16	reubidv 3126	. . . 4 |- ( ph -> ( E! x e. B ( A. y e. { X , Y } x .<_ y /\ A. z e. B ( A. y e. { X , Y } z .<_ y -> z .<_ x ) ) <-> E! x e. B ( ( x .<_ X /\ x .<_ Y ) /\ A. z e. B ( ( z .<_ X /\ z .<_ Y ) -> z .<_ x ) ) ) )
18	14, 17	sylibd 229	. . 3 |- ( ph -> ( { X , Y } e. dom ( glb ` K ) -> E! x e. B ( ( x .<_ X /\ x .<_ Y ) /\ A. z e. B ( ( z .<_ X /\ z .<_ Y ) -> z .<_ x ) ) ) )
19	7, 18	sylbid 230	. 2 |- ( ph -> ( <. X , Y >. e. dom ./\ -> E! x e. B ( ( x .<_ X /\ x .<_ Y ) /\ A. z e. B ( ( z .<_ X /\ z .<_ Y ) -> z .<_ x ) ) ) )
20	1, 19	mpd 15	1 |- ( ph -> E! x e. B ( ( x .<_ X /\ x .<_ Y ) /\ A. z e. B ( ( z .<_ X /\ z .<_ Y ) -> z .<_ x ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E!wreu 2914   {cpr 4179   <.cop 4183   class class class wbr 4653   dom cdm 5114   ` cfv 5888   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:  meetlem  17025