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Theorem suplocexprlemub 7524
Description: Lemma for suplocexpr 7526. 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 7523 . . . . . . 7  |-  ( ph  ->  B  e.  P. )
76ad2antrr 479 . . . . . 6  |-  ( ( ( ph  /\  y  e.  A )  /\  B  <P  y )  ->  B  e.  P. )
82, 3, 4suplocexprlemss 7516 . . . . . . . 8  |-  ( ph  ->  A  C_  P. )
98ad2antrr 479 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  A )  /\  B  <P  y )  ->  A  C_ 
P. )
10 simplr 519 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  A )  /\  B  <P  y )  ->  y  e.  A )
119, 10sseldd 3093 . . . . . 6  |-  ( ( ( ph  /\  y  e.  A )  /\  B  <P  y )  ->  y  e.  P. )
12 ltdfpr 7307 . . . . . 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 408 . . . . 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 528 . . . . . . . 8  |-  ( ( ( ( ph  /\  y  e.  A )  /\  B  <P  y )  /\  ( s  e. 
Q.  /\  ( s  e.  ( 2nd `  B
)  /\  s  e.  ( 1st `  y ) ) ) )  -> 
s  e.  ( 2nd `  B ) )
165suplocexprlem2b 7515 . . . . . . . . . . 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 2207 . . . . . . . . 9  |-  ( ph  ->  ( s  e.  ( 2nd `  B )  <-> 
s  e.  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } ) )
1918ad3antrrr 483 . . . . . . . 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 3928 . . . . . . . . 9  |-  ( u  =  s  ->  (
w  <Q  u  <->  w  <Q  s ) )
2221rexbidv 2436 . . . . . . . 8  |-  ( u  =  s  ->  ( E. w  e.  |^| ( 2nd " A ) w 
<Q  u  <->  E. w  e.  |^| ( 2nd " A ) w  <Q  s )
)
2322elrab 2835 . . . . . . 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 529 . . . . . . . 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 521 . . . . . . . 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 479 . . . . . . . . . 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 7276 . . . . . . . . . 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 2201 . . . . . . . . . 10  |-  ( t  =  ( 2nd `  y
)  ->  ( w  e.  t  <->  w  e.  ( 2nd `  y ) ) )
33 simprl 520 . . . . . . . . . . 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 2684 . . . . . . . . . . . 12  |-  w  e. 
_V
3534elint2 3773 . . . . . . . . . . 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 6049 . . . . . . . . . . . . 13  |-  2nd : _V -onto-> _V
38 fofun 5341 . . . . . . . . . . . . 13  |-  ( 2nd
: _V -onto-> _V  ->  Fun 
2nd )
3937, 38ax-mp 5 . . . . . . . . . . . 12  |-  Fun  2nd
40 vex 2684 . . . . . . . . . . . . 13  |-  y  e. 
_V
41 fof 5340 . . . . . . . . . . . . . . 15  |-  ( 2nd
: _V -onto-> _V  ->  2nd
: _V --> _V )
4237, 41ax-mp 5 . . . . . . . . . . . . . 14  |-  2nd : _V
--> _V
4342fdmi 5275 . . . . . . . . . . . . 13  |-  dom  2nd  =  _V
4440, 43eleqtrri 2213 . . . . . . . . . . . 12  |-  y  e. 
dom  2nd
45 funfvima 5642 . . . . . . . . . . . 12  |-  ( ( Fun  2nd  /\  y  e.  dom  2nd )  -> 
( y  e.  A  ->  ( 2nd `  y
)  e.  ( 2nd " A ) ) )
4639, 44, 45mp2an 422 . . . . . . . . . . 11  |-  ( y  e.  A  ->  ( 2nd `  y )  e.  ( 2nd " A
) )
4746ad4antlr 486 . . . . . . . . . 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 2789 . . . . . . . . 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 7286 . . . . . . . . 9  |-  ( (
<. ( 1st `  y
) ,  ( 2nd `  y ) >.  e.  P.  /\  w  e.  ( 2nd `  y ) )  -> 
( w  <Q  s  ->  s  e.  ( 2nd `  y ) ) )
5031, 48, 49syl2anc 408 . . . . . . . 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 524 . . . . . . 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 7293 . . . . . . 7  |-  ( (
<. ( 1st `  y
) ,  ( 2nd `  y ) >.  e.  P.  /\  s  e.  Q. )  ->  -.  ( s  e.  ( 1st `  y
)  /\  s  e.  ( 2nd `  y ) ) )
5531, 53, 54syl2anc 408 . . . . . 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 1351 . . . . 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 2552 . . . 4  |-  ( ( ( ( ph  /\  y  e.  A )  /\  B  <P  y )  /\  ( s  e. 
Q.  /\  ( s  e.  ( 2nd `  B
)  /\  s  e.  ( 1st `  y ) ) ) )  -> F.  )
5814, 57rexlimddv 2552 . . 3  |-  ( ( ( ph  /\  y  e.  A )  /\  B  <P  y )  -> F.  )
5958inegd 1350 . 2  |-  ( (
ph  /\  y  e.  A )  ->  -.  B  <P  y )
6059ralrimiva 2503 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 697    = wceq 1331   F. wfal 1336   E.wex 1468    e. wcel 1480   A.wral 2414   E.wrex 2415   {crab 2418   _Vcvv 2681    C_ wss 3066   <.cop 3525   U.cuni 3731   |^|cint 3766   class class class wbr 3924   dom cdm 4534   "cima 4537   Fun wfun 5112   -->wf 5114   -onto->wfo 5116   ` cfv 5118   1stc1st 6029   2ndc2nd 6030   Q.cnq 7081    <Q cltq 7086   P.cnp 7092    <P cltp 7096
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-2o 6307  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-lti 7108  df-plpq 7145  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-plqqs 7150  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-enq0 7225  df-nq0 7226  df-0nq0 7227  df-plq0 7228  df-mq0 7229  df-inp 7267  df-iltp 7271
This theorem is referenced by:  suplocexpr  7526
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