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Theorem suplocexprlemub 7685
Description: Lemma for suplocexpr 7687. The putative supremum is an upper bound. (Contributed by Jim Kingdon, 14-Jan-2024.)
Hypotheses
Ref Expression
suplocexpr.m  |-  ( ph  ->  E. x  x  e.  A )
suplocexpr.ub  |-  ( ph  ->  E. x  e.  P.  A. y  e.  A  y 
<P  x )
suplocexpr.loc  |-  ( ph  ->  A. x  e.  P.  A. y  e.  P.  (
x  <P  y  ->  ( E. z  e.  A  x  <P  z  \/  A. z  e.  A  z  <P  y ) ) )
suplocexpr.b  |-  B  = 
<. U. ( 1st " A
) ,  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } >.
Assertion
Ref Expression
suplocexprlemub  |-  ( ph  ->  A. y  e.  A  -.  B  <P  y )
Distinct variable groups:    u, A, w, y    x, A, z, u, y    w, B    ph, u, w, y    ph, x, z    z, w
Allowed substitution hints:    B( x, y, z, u)

Proof of Theorem suplocexprlemub
Dummy variables  s  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 109 . . . . 5  |-  ( ( ( ph  /\  y  e.  A )  /\  B  <P  y )  ->  B  <P  y )
2 suplocexpr.m . . . . . . . 8  |-  ( ph  ->  E. x  x  e.  A )
3 suplocexpr.ub . . . . . . . 8  |-  ( ph  ->  E. x  e.  P.  A. y  e.  A  y 
<P  x )
4 suplocexpr.loc . . . . . . . 8  |-  ( ph  ->  A. x  e.  P.  A. y  e.  P.  (
x  <P  y  ->  ( E. z  e.  A  x  <P  z  \/  A. z  e.  A  z  <P  y ) ) )
5 suplocexpr.b . . . . . . . 8  |-  B  = 
<. U. ( 1st " A
) ,  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } >.
62, 3, 4, 5suplocexprlemex 7684 . . . . . . 7  |-  ( ph  ->  B  e.  P. )
76ad2antrr 485 . . . . . 6  |-  ( ( ( ph  /\  y  e.  A )  /\  B  <P  y )  ->  B  e.  P. )
82, 3, 4suplocexprlemss 7677 . . . . . . . 8  |-  ( ph  ->  A  C_  P. )
98ad2antrr 485 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  A )  /\  B  <P  y )  ->  A  C_ 
P. )
10 simplr 525 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  A )  /\  B  <P  y )  ->  y  e.  A )
119, 10sseldd 3148 . . . . . 6  |-  ( ( ( ph  /\  y  e.  A )  /\  B  <P  y )  ->  y  e.  P. )
12 ltdfpr 7468 . . . . . 6  |-  ( ( B  e.  P.  /\  y  e.  P. )  ->  ( B  <P  y  <->  E. s  e.  Q.  (
s  e.  ( 2nd `  B )  /\  s  e.  ( 1st `  y
) ) ) )
137, 11, 12syl2anc 409 . . . . 5  |-  ( ( ( ph  /\  y  e.  A )  /\  B  <P  y )  ->  ( B  <P  y  <->  E. s  e.  Q.  ( s  e.  ( 2nd `  B
)  /\  s  e.  ( 1st `  y ) ) ) )
141, 13mpbid 146 . . . 4  |-  ( ( ( ph  /\  y  e.  A )  /\  B  <P  y )  ->  E. s  e.  Q.  ( s  e.  ( 2nd `  B
)  /\  s  e.  ( 1st `  y ) ) )
15 simprrl 534 . . . . . . . 8  |-  ( ( ( ( ph  /\  y  e.  A )  /\  B  <P  y )  /\  ( s  e. 
Q.  /\  ( s  e.  ( 2nd `  B
)  /\  s  e.  ( 1st `  y ) ) ) )  -> 
s  e.  ( 2nd `  B ) )
165suplocexprlem2b 7676 . . . . . . . . . . 11  |-  ( A 
C_  P.  ->  ( 2nd `  B )  =  {
u  e.  Q.  |  E. w  e.  |^| ( 2nd " A ) w 
<Q  u } )
178, 16syl 14 . . . . . . . . . 10  |-  ( ph  ->  ( 2nd `  B
)  =  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } )
1817eleq2d 2240 . . . . . . . . 9  |-  ( ph  ->  ( s  e.  ( 2nd `  B )  <-> 
s  e.  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } ) )
1918ad3antrrr 489 . . . . . . . 8  |-  ( ( ( ( ph  /\  y  e.  A )  /\  B  <P  y )  /\  ( s  e. 
Q.  /\  ( s  e.  ( 2nd `  B
)  /\  s  e.  ( 1st `  y ) ) ) )  -> 
( s  e.  ( 2nd `  B )  <-> 
s  e.  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } ) )
2015, 19mpbid 146 . . . . . . 7  |-  ( ( ( ( ph  /\  y  e.  A )  /\  B  <P  y )  /\  ( s  e. 
Q.  /\  ( s  e.  ( 2nd `  B
)  /\  s  e.  ( 1st `  y ) ) ) )  -> 
s  e.  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } )
21 breq2 3993 . . . . . . . . 9  |-  ( u  =  s  ->  (
w  <Q  u  <->  w  <Q  s ) )
2221rexbidv 2471 . . . . . . . 8  |-  ( u  =  s  ->  ( E. w  e.  |^| ( 2nd " A ) w 
<Q  u  <->  E. w  e.  |^| ( 2nd " A ) w  <Q  s )
)
2322elrab 2886 . . . . . . 7  |-  ( s  e.  { u  e. 
Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } 
<->  ( s  e.  Q.  /\ 
E. w  e.  |^| ( 2nd " A ) w  <Q  s )
)
2420, 23sylib 121 . . . . . 6  |-  ( ( ( ( ph  /\  y  e.  A )  /\  B  <P  y )  /\  ( s  e. 
Q.  /\  ( s  e.  ( 2nd `  B
)  /\  s  e.  ( 1st `  y ) ) ) )  -> 
( s  e.  Q.  /\ 
E. w  e.  |^| ( 2nd " A ) w  <Q  s )
)
2524simprd 113 . . . . 5  |-  ( ( ( ( ph  /\  y  e.  A )  /\  B  <P  y )  /\  ( s  e. 
Q.  /\  ( s  e.  ( 2nd `  B
)  /\  s  e.  ( 1st `  y ) ) ) )  ->  E. w  e.  |^| ( 2nd " A ) w 
<Q  s )
26 simprrr 535 . . . . . . . 8  |-  ( ( ( ( ph  /\  y  e.  A )  /\  B  <P  y )  /\  ( s  e. 
Q.  /\  ( s  e.  ( 2nd `  B
)  /\  s  e.  ( 1st `  y ) ) ) )  -> 
s  e.  ( 1st `  y ) )
2726adantr 274 . . . . . . 7  |-  ( ( ( ( ( ph  /\  y  e.  A )  /\  B  <P  y
)  /\  ( s  e.  Q.  /\  ( s  e.  ( 2nd `  B
)  /\  s  e.  ( 1st `  y ) ) ) )  /\  ( w  e.  |^| ( 2nd " A )  /\  w  <Q  s ) )  ->  s  e.  ( 1st `  y ) )
28 simprr 527 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  y  e.  A )  /\  B  <P  y
)  /\  ( s  e.  Q.  /\  ( s  e.  ( 2nd `  B
)  /\  s  e.  ( 1st `  y ) ) ) )  /\  ( w  e.  |^| ( 2nd " A )  /\  w  <Q  s ) )  ->  w  <Q  s
)
2911ad2antrr 485 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  y  e.  A )  /\  B  <P  y
)  /\  ( s  e.  Q.  /\  ( s  e.  ( 2nd `  B
)  /\  s  e.  ( 1st `  y ) ) ) )  /\  ( w  e.  |^| ( 2nd " A )  /\  w  <Q  s ) )  ->  y  e.  P. )
30 prop 7437 . . . . . . . . . 10  |-  ( y  e.  P.  ->  <. ( 1st `  y ) ,  ( 2nd `  y
) >.  e.  P. )
3129, 30syl 14 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  y  e.  A )  /\  B  <P  y
)  /\  ( s  e.  Q.  /\  ( s  e.  ( 2nd `  B
)  /\  s  e.  ( 1st `  y ) ) ) )  /\  ( w  e.  |^| ( 2nd " A )  /\  w  <Q  s ) )  ->  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  e.  P. )
32 eleq2 2234 . . . . . . . . . 10  |-  ( t  =  ( 2nd `  y
)  ->  ( w  e.  t  <->  w  e.  ( 2nd `  y ) ) )
33 simprl 526 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  y  e.  A )  /\  B  <P  y
)  /\  ( s  e.  Q.  /\  ( s  e.  ( 2nd `  B
)  /\  s  e.  ( 1st `  y ) ) ) )  /\  ( w  e.  |^| ( 2nd " A )  /\  w  <Q  s ) )  ->  w  e.  |^| ( 2nd " A ) )
34 vex 2733 . . . . . . . . . . . 12  |-  w  e. 
_V
3534elint2 3838 . . . . . . . . . . 11  |-  ( w  e.  |^| ( 2nd " A
)  <->  A. t  e.  ( 2nd " A ) w  e.  t )
3633, 35sylib 121 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  y  e.  A )  /\  B  <P  y
)  /\  ( s  e.  Q.  /\  ( s  e.  ( 2nd `  B
)  /\  s  e.  ( 1st `  y ) ) ) )  /\  ( w  e.  |^| ( 2nd " A )  /\  w  <Q  s ) )  ->  A. t  e.  ( 2nd " A ) w  e.  t )
37 fo2nd 6137 . . . . . . . . . . . . 13  |-  2nd : _V -onto-> _V
38 fofun 5421 . . . . . . . . . . . . 13  |-  ( 2nd
: _V -onto-> _V  ->  Fun 
2nd )
3937, 38ax-mp 5 . . . . . . . . . . . 12  |-  Fun  2nd
40 vex 2733 . . . . . . . . . . . . 13  |-  y  e. 
_V
41 fof 5420 . . . . . . . . . . . . . . 15  |-  ( 2nd
: _V -onto-> _V  ->  2nd
: _V --> _V )
4237, 41ax-mp 5 . . . . . . . . . . . . . 14  |-  2nd : _V
--> _V
4342fdmi 5355 . . . . . . . . . . . . 13  |-  dom  2nd  =  _V
4440, 43eleqtrri 2246 . . . . . . . . . . . 12  |-  y  e. 
dom  2nd
45 funfvima 5727 . . . . . . . . . . . 12  |-  ( ( Fun  2nd  /\  y  e.  dom  2nd )  -> 
( y  e.  A  ->  ( 2nd `  y
)  e.  ( 2nd " A ) ) )
4639, 44, 45mp2an 424 . . . . . . . . . . 11  |-  ( y  e.  A  ->  ( 2nd `  y )  e.  ( 2nd " A
) )
4746ad4antlr 492 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  y  e.  A )  /\  B  <P  y
)  /\  ( s  e.  Q.  /\  ( s  e.  ( 2nd `  B
)  /\  s  e.  ( 1st `  y ) ) ) )  /\  ( w  e.  |^| ( 2nd " A )  /\  w  <Q  s ) )  ->  ( 2nd `  y
)  e.  ( 2nd " A ) )
4832, 36, 47rspcdva 2839 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  y  e.  A )  /\  B  <P  y
)  /\  ( s  e.  Q.  /\  ( s  e.  ( 2nd `  B
)  /\  s  e.  ( 1st `  y ) ) ) )  /\  ( w  e.  |^| ( 2nd " A )  /\  w  <Q  s ) )  ->  w  e.  ( 2nd `  y ) )
49 prcunqu 7447 . . . . . . . . 9  |-  ( (
<. ( 1st `  y
) ,  ( 2nd `  y ) >.  e.  P.  /\  w  e.  ( 2nd `  y ) )  -> 
( w  <Q  s  ->  s  e.  ( 2nd `  y ) ) )
5031, 48, 49syl2anc 409 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  y  e.  A )  /\  B  <P  y
)  /\  ( s  e.  Q.  /\  ( s  e.  ( 2nd `  B
)  /\  s  e.  ( 1st `  y ) ) ) )  /\  ( w  e.  |^| ( 2nd " A )  /\  w  <Q  s ) )  ->  ( w  <Q  s  ->  s  e.  ( 2nd `  y ) ) )
5128, 50mpd 13 . . . . . . 7  |-  ( ( ( ( ( ph  /\  y  e.  A )  /\  B  <P  y
)  /\  ( s  e.  Q.  /\  ( s  e.  ( 2nd `  B
)  /\  s  e.  ( 1st `  y ) ) ) )  /\  ( w  e.  |^| ( 2nd " A )  /\  w  <Q  s ) )  ->  s  e.  ( 2nd `  y ) )
5227, 51jca 304 . . . . . 6  |-  ( ( ( ( ( ph  /\  y  e.  A )  /\  B  <P  y
)  /\  ( s  e.  Q.  /\  ( s  e.  ( 2nd `  B
)  /\  s  e.  ( 1st `  y ) ) ) )  /\  ( w  e.  |^| ( 2nd " A )  /\  w  <Q  s ) )  ->  ( s  e.  ( 1st `  y
)  /\  s  e.  ( 2nd `  y ) ) )
53 simplrl 530 . . . . . . 7  |-  ( ( ( ( ( ph  /\  y  e.  A )  /\  B  <P  y
)  /\  ( s  e.  Q.  /\  ( s  e.  ( 2nd `  B
)  /\  s  e.  ( 1st `  y ) ) ) )  /\  ( w  e.  |^| ( 2nd " A )  /\  w  <Q  s ) )  ->  s  e.  Q. )
54 prdisj 7454 . . . . . . 7  |-  ( (
<. ( 1st `  y
) ,  ( 2nd `  y ) >.  e.  P.  /\  s  e.  Q. )  ->  -.  ( s  e.  ( 1st `  y
)  /\  s  e.  ( 2nd `  y ) ) )
5531, 53, 54syl2anc 409 . . . . . 6  |-  ( ( ( ( ( ph  /\  y  e.  A )  /\  B  <P  y
)  /\  ( s  e.  Q.  /\  ( s  e.  ( 2nd `  B
)  /\  s  e.  ( 1st `  y ) ) ) )  /\  ( w  e.  |^| ( 2nd " A )  /\  w  <Q  s ) )  ->  -.  ( s  e.  ( 1st `  y
)  /\  s  e.  ( 2nd `  y ) ) )
5652, 55pm2.21fal 1368 . . . . 5  |-  ( ( ( ( ( ph  /\  y  e.  A )  /\  B  <P  y
)  /\  ( s  e.  Q.  /\  ( s  e.  ( 2nd `  B
)  /\  s  e.  ( 1st `  y ) ) ) )  /\  ( w  e.  |^| ( 2nd " A )  /\  w  <Q  s ) )  -> F.  )
5725, 56rexlimddv 2592 . . . 4  |-  ( ( ( ( ph  /\  y  e.  A )  /\  B  <P  y )  /\  ( s  e. 
Q.  /\  ( s  e.  ( 2nd `  B
)  /\  s  e.  ( 1st `  y ) ) ) )  -> F.  )
5814, 57rexlimddv 2592 . . 3  |-  ( ( ( ph  /\  y  e.  A )  /\  B  <P  y )  -> F.  )
5958inegd 1367 . 2  |-  ( (
ph  /\  y  e.  A )  ->  -.  B  <P  y )
6059ralrimiva 2543 1  |-  ( ph  ->  A. y  e.  A  -.  B  <P  y )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 703    = wceq 1348   F. wfal 1353   E.wex 1485    e. wcel 2141   A.wral 2448   E.wrex 2449   {crab 2452   _Vcvv 2730    C_ wss 3121   <.cop 3586   U.cuni 3796   |^|cint 3831   class class class wbr 3989   dom cdm 4611   "cima 4614   Fun wfun 5192   -->wf 5194   -onto->wfo 5196   ` cfv 5198   1stc1st 6117   2ndc2nd 6118   Q.cnq 7242    <Q cltq 7247   P.cnp 7253    <P cltp 7257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iltp 7432
This theorem is referenced by:  suplocexpr  7687
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