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Theorem suplocexprlemru 7520
Description: Lemma for suplocexpr 7526. The upper cut of the putative supremum is rounded. (Contributed by Jim Kingdon, 9-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
suplocexprlemru  |-  ( ph  ->  A. r  e.  Q.  ( r  e.  ( 2nd `  B )  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  B ) ) ) )
Distinct variable groups:    A, q, u   
x, A, y    B, q, w    ph, q, r, w    ph, x, y    u, r, w
Allowed substitution hints:    ph( z, u)    A( z, w, r)    B( x, y, z, u, r)

Proof of Theorem suplocexprlemru
StepHypRef Expression
1 suplocexpr.m . . . . . . . . . . . 12  |-  ( ph  ->  E. x  x  e.  A )
2 suplocexpr.ub . . . . . . . . . . . 12  |-  ( ph  ->  E. x  e.  P.  A. y  e.  A  y 
<P  x )
3 suplocexpr.loc . . . . . . . . . . . 12  |-  ( 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 ) ) )
41, 2, 3suplocexprlemss 7516 . . . . . . . . . . 11  |-  ( ph  ->  A  C_  P. )
5 suplocexpr.b . . . . . . . . . . . 12  |-  B  = 
<. U. ( 1st " A
) ,  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } >.
65suplocexprlem2b 7515 . . . . . . . . . . 11  |-  ( A 
C_  P.  ->  ( 2nd `  B )  =  {
u  e.  Q.  |  E. w  e.  |^| ( 2nd " A ) w 
<Q  u } )
74, 6syl 14 . . . . . . . . . 10  |-  ( ph  ->  ( 2nd `  B
)  =  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } )
87eleq2d 2207 . . . . . . . . 9  |-  ( ph  ->  ( r  e.  ( 2nd `  B )  <-> 
r  e.  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } ) )
98adantr 274 . . . . . . . 8  |-  ( (
ph  /\  r  e.  Q. )  ->  ( r  e.  ( 2nd `  B
)  <->  r  e.  {
u  e.  Q.  |  E. w  e.  |^| ( 2nd " A ) w 
<Q  u } ) )
109biimpa 294 . . . . . . 7  |-  ( ( ( ph  /\  r  e.  Q. )  /\  r  e.  ( 2nd `  B
) )  ->  r  e.  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A ) w  <Q  u }
)
11 breq2 3928 . . . . . . . . 9  |-  ( u  =  r  ->  (
w  <Q  u  <->  w  <Q  r ) )
1211rexbidv 2436 . . . . . . . 8  |-  ( u  =  r  ->  ( E. w  e.  |^| ( 2nd " A ) w 
<Q  u  <->  E. w  e.  |^| ( 2nd " A ) w  <Q  r )
)
1312elrab 2835 . . . . . . 7  |-  ( r  e.  { u  e. 
Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } 
<->  ( r  e.  Q.  /\ 
E. w  e.  |^| ( 2nd " A ) w  <Q  r )
)
1410, 13sylib 121 . . . . . 6  |-  ( ( ( ph  /\  r  e.  Q. )  /\  r  e.  ( 2nd `  B
) )  ->  (
r  e.  Q.  /\  E. w  e.  |^| ( 2nd " A ) w 
<Q  r ) )
1514simprd 113 . . . . 5  |-  ( ( ( ph  /\  r  e.  Q. )  /\  r  e.  ( 2nd `  B
) )  ->  E. w  e.  |^| ( 2nd " A
) w  <Q  r
)
16 ltbtwnnqq 7216 . . . . . . . 8  |-  ( w 
<Q  r  <->  E. q  e.  Q.  ( w  <Q  q  /\  q  <Q  r ) )
1716biimpi 119 . . . . . . 7  |-  ( w 
<Q  r  ->  E. q  e.  Q.  ( w  <Q  q  /\  q  <Q  r
) )
1817ad2antll 482 . . . . . 6  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  r  e.  ( 2nd `  B ) )  /\  ( w  e. 
|^| ( 2nd " A
)  /\  w  <Q  r ) )  ->  E. q  e.  Q.  ( w  <Q  q  /\  q  <Q  r
) )
19 simprr 521 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  r  e.  Q. )  /\  r  e.  ( 2nd `  B
) )  /\  (
w  e.  |^| ( 2nd " A )  /\  w  <Q  r ) )  /\  q  e.  Q. )  /\  ( w  <Q  q  /\  q  <Q  r
) )  ->  q  <Q  r )
20 breq2 3928 . . . . . . . . . . . 12  |-  ( u  =  q  ->  (
w  <Q  u  <->  w  <Q  q ) )
2120rexbidv 2436 . . . . . . . . . . 11  |-  ( u  =  q  ->  ( E. w  e.  |^| ( 2nd " A ) w 
<Q  u  <->  E. w  e.  |^| ( 2nd " A ) w  <Q  q )
)
22 simplr 519 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  r  e.  Q. )  /\  r  e.  ( 2nd `  B
) )  /\  (
w  e.  |^| ( 2nd " A )  /\  w  <Q  r ) )  /\  q  e.  Q. )  /\  ( w  <Q  q  /\  q  <Q  r
) )  ->  q  e.  Q. )
23 simprl 520 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  r  e.  ( 2nd `  B ) )  /\  ( w  e. 
|^| ( 2nd " A
)  /\  w  <Q  r ) )  ->  w  e.  |^| ( 2nd " A
) )
2423ad2antrr 479 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  r  e.  Q. )  /\  r  e.  ( 2nd `  B
) )  /\  (
w  e.  |^| ( 2nd " A )  /\  w  <Q  r ) )  /\  q  e.  Q. )  /\  ( w  <Q  q  /\  q  <Q  r
) )  ->  w  e.  |^| ( 2nd " A
) )
25 simprl 520 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  r  e.  Q. )  /\  r  e.  ( 2nd `  B
) )  /\  (
w  e.  |^| ( 2nd " A )  /\  w  <Q  r ) )  /\  q  e.  Q. )  /\  ( w  <Q  q  /\  q  <Q  r
) )  ->  w  <Q  q )
2624, 25jca 304 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  r  e.  Q. )  /\  r  e.  ( 2nd `  B
) )  /\  (
w  e.  |^| ( 2nd " A )  /\  w  <Q  r ) )  /\  q  e.  Q. )  /\  ( w  <Q  q  /\  q  <Q  r
) )  ->  (
w  e.  |^| ( 2nd " A )  /\  w  <Q  q ) )
27 rspe 2479 . . . . . . . . . . . 12  |-  ( ( w  e.  |^| ( 2nd " A )  /\  w  <Q  q )  ->  E. w  e.  |^| ( 2nd " A ) w 
<Q  q )
2826, 27syl 14 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  r  e.  Q. )  /\  r  e.  ( 2nd `  B
) )  /\  (
w  e.  |^| ( 2nd " A )  /\  w  <Q  r ) )  /\  q  e.  Q. )  /\  ( w  <Q  q  /\  q  <Q  r
) )  ->  E. w  e.  |^| ( 2nd " A
) w  <Q  q
)
2921, 22, 28elrabd 2837 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  r  e.  Q. )  /\  r  e.  ( 2nd `  B
) )  /\  (
w  e.  |^| ( 2nd " A )  /\  w  <Q  r ) )  /\  q  e.  Q. )  /\  ( w  <Q  q  /\  q  <Q  r
) )  ->  q  e.  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A ) w  <Q  u }
)
307eleq2d 2207 . . . . . . . . . . 11  |-  ( ph  ->  ( q  e.  ( 2nd `  B )  <-> 
q  e.  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } ) )
3130ad5antr 487 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  r  e.  Q. )  /\  r  e.  ( 2nd `  B
) )  /\  (
w  e.  |^| ( 2nd " A )  /\  w  <Q  r ) )  /\  q  e.  Q. )  /\  ( w  <Q  q  /\  q  <Q  r
) )  ->  (
q  e.  ( 2nd `  B )  <->  q  e.  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A ) w 
<Q  u } ) )
3229, 31mpbird 166 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  r  e.  Q. )  /\  r  e.  ( 2nd `  B
) )  /\  (
w  e.  |^| ( 2nd " A )  /\  w  <Q  r ) )  /\  q  e.  Q. )  /\  ( w  <Q  q  /\  q  <Q  r
) )  ->  q  e.  ( 2nd `  B
) )
3319, 32jca 304 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  r  e.  Q. )  /\  r  e.  ( 2nd `  B
) )  /\  (
w  e.  |^| ( 2nd " A )  /\  w  <Q  r ) )  /\  q  e.  Q. )  /\  ( w  <Q  q  /\  q  <Q  r
) )  ->  (
q  <Q  r  /\  q  e.  ( 2nd `  B
) ) )
3433ex 114 . . . . . . 7  |-  ( ( ( ( ( ph  /\  r  e.  Q. )  /\  r  e.  ( 2nd `  B ) )  /\  ( w  e. 
|^| ( 2nd " A
)  /\  w  <Q  r ) )  /\  q  e.  Q. )  ->  (
( w  <Q  q  /\  q  <Q  r )  ->  ( q  <Q 
r  /\  q  e.  ( 2nd `  B ) ) ) )
3534reximdva 2532 . . . . . 6  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  r  e.  ( 2nd `  B ) )  /\  ( w  e. 
|^| ( 2nd " A
)  /\  w  <Q  r ) )  ->  ( E. q  e.  Q.  ( w  <Q  q  /\  q  <Q  r )  ->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  B ) ) ) )
3618, 35mpd 13 . . . . 5  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  r  e.  ( 2nd `  B ) )  /\  ( w  e. 
|^| ( 2nd " A
)  /\  w  <Q  r ) )  ->  E. q  e.  Q.  ( q  <Q 
r  /\  q  e.  ( 2nd `  B ) ) )
3715, 36rexlimddv 2552 . . . 4  |-  ( ( ( ph  /\  r  e.  Q. )  /\  r  e.  ( 2nd `  B
) )  ->  E. q  e.  Q.  ( q  <Q 
r  /\  q  e.  ( 2nd `  B ) ) )
3837ex 114 . . 3  |-  ( (
ph  /\  r  e.  Q. )  ->  ( r  e.  ( 2nd `  B
)  ->  E. q  e.  Q.  ( q  <Q 
r  /\  q  e.  ( 2nd `  B ) ) ) )
39 simpllr 523 . . . . . 6  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  q  e.  Q. )  /\  ( q  <Q 
r  /\  q  e.  ( 2nd `  B ) ) )  ->  r  e.  Q. )
40 simprr 521 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  q  e.  Q. )  /\  ( q  <Q 
r  /\  q  e.  ( 2nd `  B ) ) )  ->  q  e.  ( 2nd `  B
) )
4130ad3antrrr 483 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  q  e.  Q. )  /\  ( q  <Q 
r  /\  q  e.  ( 2nd `  B ) ) )  ->  (
q  e.  ( 2nd `  B )  <->  q  e.  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A ) w 
<Q  u } ) )
4240, 41mpbid 146 . . . . . . . . 9  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  q  e.  Q. )  /\  ( q  <Q 
r  /\  q  e.  ( 2nd `  B ) ) )  ->  q  e.  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A ) w  <Q  u }
)
4321elrab 2835 . . . . . . . . 9  |-  ( q  e.  { u  e. 
Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } 
<->  ( q  e.  Q.  /\ 
E. w  e.  |^| ( 2nd " A ) w  <Q  q )
)
4442, 43sylib 121 . . . . . . . 8  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  q  e.  Q. )  /\  ( q  <Q 
r  /\  q  e.  ( 2nd `  B ) ) )  ->  (
q  e.  Q.  /\  E. w  e.  |^| ( 2nd " A ) w 
<Q  q ) )
4544simprd 113 . . . . . . 7  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  q  e.  Q. )  /\  ( q  <Q 
r  /\  q  e.  ( 2nd `  B ) ) )  ->  E. w  e.  |^| ( 2nd " A
) w  <Q  q
)
46 simpr 109 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  r  e.  Q. )  /\  q  e.  Q. )  /\  (
q  <Q  r  /\  q  e.  ( 2nd `  B
) ) )  /\  w  e.  |^| ( 2nd " A ) )  /\  w  <Q  q )  ->  w  <Q  q )
47 simprl 520 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  q  e.  Q. )  /\  ( q  <Q 
r  /\  q  e.  ( 2nd `  B ) ) )  ->  q  <Q  r )
4847ad2antrr 479 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  r  e.  Q. )  /\  q  e.  Q. )  /\  (
q  <Q  r  /\  q  e.  ( 2nd `  B
) ) )  /\  w  e.  |^| ( 2nd " A ) )  /\  w  <Q  q )  -> 
q  <Q  r )
4946, 48jca 304 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  r  e.  Q. )  /\  q  e.  Q. )  /\  (
q  <Q  r  /\  q  e.  ( 2nd `  B
) ) )  /\  w  e.  |^| ( 2nd " A ) )  /\  w  <Q  q )  -> 
( w  <Q  q  /\  q  <Q  r ) )
50 ltrelnq 7166 . . . . . . . . . . . . . 14  |-  <Q  C_  ( Q.  X.  Q. )
5150brel 4586 . . . . . . . . . . . . 13  |-  ( w 
<Q  q  ->  ( w  e.  Q.  /\  q  e.  Q. ) )
5251simpld 111 . . . . . . . . . . . 12  |-  ( w 
<Q  q  ->  w  e. 
Q. )
5352adantl 275 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  r  e.  Q. )  /\  q  e.  Q. )  /\  (
q  <Q  r  /\  q  e.  ( 2nd `  B
) ) )  /\  w  e.  |^| ( 2nd " A ) )  /\  w  <Q  q )  ->  w  e.  Q. )
54 simp-4r 531 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  r  e.  Q. )  /\  q  e.  Q. )  /\  (
q  <Q  r  /\  q  e.  ( 2nd `  B
) ) )  /\  w  e.  |^| ( 2nd " A ) )  /\  w  <Q  q )  -> 
q  e.  Q. )
5539ad2antrr 479 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  r  e.  Q. )  /\  q  e.  Q. )  /\  (
q  <Q  r  /\  q  e.  ( 2nd `  B
) ) )  /\  w  e.  |^| ( 2nd " A ) )  /\  w  <Q  q )  -> 
r  e.  Q. )
56 ltsonq 7199 . . . . . . . . . . . 12  |-  <Q  Or  Q.
57 sotr 4235 . . . . . . . . . . . 12  |-  ( ( 
<Q  Or  Q.  /\  (
w  e.  Q.  /\  q  e.  Q.  /\  r  e.  Q. ) )  -> 
( ( w  <Q  q  /\  q  <Q  r
)  ->  w  <Q  r ) )
5856, 57mpan 420 . . . . . . . . . . 11  |-  ( ( w  e.  Q.  /\  q  e.  Q.  /\  r  e.  Q. )  ->  (
( w  <Q  q  /\  q  <Q  r )  ->  w  <Q  r
) )
5953, 54, 55, 58syl3anc 1216 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  r  e.  Q. )  /\  q  e.  Q. )  /\  (
q  <Q  r  /\  q  e.  ( 2nd `  B
) ) )  /\  w  e.  |^| ( 2nd " A ) )  /\  w  <Q  q )  -> 
( ( w  <Q  q  /\  q  <Q  r
)  ->  w  <Q  r ) )
6049, 59mpd 13 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  r  e.  Q. )  /\  q  e.  Q. )  /\  (
q  <Q  r  /\  q  e.  ( 2nd `  B
) ) )  /\  w  e.  |^| ( 2nd " A ) )  /\  w  <Q  q )  ->  w  <Q  r )
6160ex 114 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  r  e.  Q. )  /\  q  e.  Q. )  /\  ( q  <Q 
r  /\  q  e.  ( 2nd `  B ) ) )  /\  w  e.  |^| ( 2nd " A
) )  ->  (
w  <Q  q  ->  w  <Q  r ) )
6261reximdva 2532 . . . . . . 7  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  q  e.  Q. )  /\  ( q  <Q 
r  /\  q  e.  ( 2nd `  B ) ) )  ->  ( E. w  e.  |^| ( 2nd " A ) w 
<Q  q  ->  E. w  e.  |^| ( 2nd " A
) w  <Q  r
) )
6345, 62mpd 13 . . . . . 6  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  q  e.  Q. )  /\  ( q  <Q 
r  /\  q  e.  ( 2nd `  B ) ) )  ->  E. w  e.  |^| ( 2nd " A
) w  <Q  r
)
6412, 39, 63elrabd 2837 . . . . 5  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  q  e.  Q. )  /\  ( q  <Q 
r  /\  q  e.  ( 2nd `  B ) ) )  ->  r  e.  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A ) w  <Q  u }
)
658ad3antrrr 483 . . . . 5  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  q  e.  Q. )  /\  ( q  <Q 
r  /\  q  e.  ( 2nd `  B ) ) )  ->  (
r  e.  ( 2nd `  B )  <->  r  e.  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A ) w 
<Q  u } ) )
6664, 65mpbird 166 . . . 4  |-  ( ( ( ( ph  /\  r  e.  Q. )  /\  q  e.  Q. )  /\  ( q  <Q 
r  /\  q  e.  ( 2nd `  B ) ) )  ->  r  e.  ( 2nd `  B
) )
6766rexlimdva2 2550 . . 3  |-  ( (
ph  /\  r  e.  Q. )  ->  ( E. q  e.  Q.  (
q  <Q  r  /\  q  e.  ( 2nd `  B
) )  ->  r  e.  ( 2nd `  B
) ) )
6838, 67impbid 128 . 2  |-  ( (
ph  /\  r  e.  Q. )  ->  ( r  e.  ( 2nd `  B
)  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  B ) ) ) )
6968ralrimiva 2503 1  |-  ( ph  ->  A. r  e.  Q.  ( r  e.  ( 2nd `  B )  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  B ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697    /\ w3a 962    = wceq 1331   E.wex 1468    e. wcel 1480   A.wral 2414   E.wrex 2415   {crab 2418    C_ wss 3066   <.cop 3525   U.cuni 3731   |^|cint 3766   class class class wbr 3924    Or wor 4212   "cima 4537   ` 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-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-inp 7267  df-iltp 7271
This theorem is referenced by:  suplocexprlemex  7523
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