Theorem poslubdg 17149

Description: Properties which determine the least upper bound in a poset. (Contributed by Stefan O'Rear, 31-Jan-2015.)

Hypotheses

Ref	Expression
poslubdg.l	|- .<_ = ( le ` K )
poslubdg.b	|- ( ph -> B = ( Base ` K ) )
poslubdg.u	|- ( ph -> U = ( lub ` K ) )
poslubdg.k	|- ( ph -> K e. Poset )
poslubdg.s	|- ( ph -> S C_ B )
poslubdg.t	|- ( ph -> T e. B )
poslubdg.ub	|- ( ( ph /\ x e. S ) -> x .<_ T )
poslubdg.le	|- ( ( ph /\ y e. B /\ A. x e. S x .<_ y ) -> T .<_ y )

Assertion

Ref	Expression
poslubdg	|- ( ph -> ( U ` S ) = T )

Distinct variable groups:   x, .<_ , y   x, B, y   x, K, y   x, S, y   x, U, y   x, T, y   ph, x, y

Proof of Theorem poslubdg

Step	Hyp	Ref	Expression
1		poslubdg.u	. . 3 |- ( ph -> U = ( lub ` K ) )
2	1	fveq1d 6193	. 2 |- ( ph -> ( U ` S ) = ( ( lub ` K ) ` S ) )
3		poslubdg.l	. . 3 |- .<_ = ( le ` K )
4		eqid 2622	. . 3 |- ( Base ` K ) = ( Base ` K )
5		eqid 2622	. . 3 |- ( lub ` K ) = ( lub ` K )
6		poslubdg.k	. . 3 |- ( ph -> K e. Poset )
7		poslubdg.s	. . . 4 |- ( ph -> S C_ B )
8		poslubdg.b	. . . 4 |- ( ph -> B = ( Base ` K ) )
9	7, 8	sseqtrd 3641	. . 3 |- ( ph -> S C_ ( Base ` K ) )
10		poslubdg.t	. . . 4 |- ( ph -> T e. B )
11	10, 8	eleqtrd 2703	. . 3 |- ( ph -> T e. ( Base ` K ) )
12		poslubdg.ub	. . 3 |- ( ( ph /\ x e. S ) -> x .<_ T )
13	8	eleq2d 2687	. . . . . 6 |- ( ph -> ( y e. B <-> y e. ( Base ` K ) ) )
14	13	biimpar 502	. . . . 5 |- ( ( ph /\ y e. ( Base ` K ) ) -> y e. B )
15	14	3adant3 1081	. . . 4 |- ( ( ph /\ y e. ( Base ` K ) /\ A. x e. S x .<_ y ) -> y e. B )
16		poslubdg.le	. . . 4 |- ( ( ph /\ y e. B /\ A. x e. S x .<_ y ) -> T .<_ y )
17	15, 16	syld3an2 1373	. . 3 |- ( ( ph /\ y e. ( Base ` K ) /\ A. x e. S x .<_ y ) -> T .<_ y )
18	3, 4, 5, 6, 9, 11, 12, 17	poslubd 17148	. 2 |- ( ph -> ( ( lub ` K ) ` S ) = T )
19	2, 18	eqtrd 2656	1 |- ( ph -> ( U ` S ) = T )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   class class class wbr 4653   ` cfv 5888   Basecbs 15857   lecple 15948   Posetcpo 16940   lubclub 16942

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

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-rmo 2920  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-preset 16928  df-poset 16946  df-lub 16974

This theorem is referenced by:  posglbd  17150  mrelatlub  17186