Theorem lubid 16990

Description: The LUB of elements less than or equal to a fixed value equals that value. (Contributed by NM, 19-Oct-2011.) (Revised by NM, 7-Sep-2018.)

Hypotheses

Ref	Expression
lubid.b	|- B = ( Base ` K )
lubid.l	|- .<_ = ( le ` K )
lubid.u	|- U = ( lub ` K )
lubid.k	|- ( ph -> K e. Poset )
lubid.x	|- ( ph -> X e. B )

Assertion

Ref	Expression
lubid	|- ( ph -> ( U ` { y e. B | y .<_ X } ) = X )

Distinct variable groups:   y, .<_   y, B   y, X

Allowed substitution hints:   ph( y)   U( y)   K( y)

Proof of Theorem lubid

Dummy variables x w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lubid.b	. . 3 |- B = ( Base ` K )
2		lubid.l	. . 3 |- .<_ = ( le ` K )
3		lubid.u	. . 3 |- U = ( lub ` K )
4		biid 251	. . 3 |- ( ( A. z e. { y e. B | y .<_ X } z .<_ x /\ A. w e. B ( A. z e. { y e. B | y .<_ X } z .<_ w -> x .<_ w ) ) <-> ( A. z e. { y e. B | y .<_ X } z .<_ x /\ A. w e. B ( A. z e. { y e. B | y .<_ X } z .<_ w -> x .<_ w ) ) )
5		lubid.k	. . 3 |- ( ph -> K e. Poset )
6		ssrab2 3687	. . . 4 |- { y e. B | y .<_ X } C_ B
7	6	a1i 11	. . 3 |- ( ph -> { y e. B | y .<_ X } C_ B )
8	1, 2, 3, 4, 5, 7	lubval 16984	. 2 |- ( ph -> ( U ` { y e. B | y .<_ X } ) = ( iota_ x e. B ( A. z e. { y e. B | y .<_ X } z .<_ x /\ A. w e. B ( A. z e. { y e. B | y .<_ X } z .<_ w -> x .<_ w ) ) ) )
9		lubid.x	. . 3 |- ( ph -> X e. B )
10	1, 2, 3, 5, 9	lublecllem 16988	. . 3 |- ( ( ph /\ x e. B ) -> ( ( A. z e. { y e. B | y .<_ X } z .<_ x /\ A. w e. B ( A. z e. { y e. B | y .<_ X } z .<_ w -> x .<_ w ) ) <-> x = X ) )
11	9, 10	riota5 6637	. 2 |- ( ph -> ( iota_ x e. B ( A. z e. { y e. B | y .<_ X } z .<_ x /\ A. w e. B ( A. z e. { y e. B | y .<_ X } z .<_ w -> x .<_ w ) ) ) = X )
12	8, 11	eqtrd 2656	1 |- ( ph -> ( U ` { y e. B | y .<_ X } ) = X )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   C_ wss 3574   class class class wbr 4653   ` cfv 5888   iota_crio 6610   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-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:  atlatmstc  34606