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Theorem recexprlemloc 7432
Description:  B is located. Lemma for recexpr 7439. (Contributed by Jim Kingdon, 27-Dec-2019.)
Hypothesis
Ref Expression
recexpr.1  |-  B  = 
<. { x  |  E. y ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  {
x  |  E. y
( y  <Q  x  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) }
>.
Assertion
Ref Expression
recexprlemloc  |-  ( A  e.  P.  ->  A. q  e.  Q.  A. r  e. 
Q.  ( q  <Q 
r  ->  ( q  e.  ( 1st `  B
)  \/  r  e.  ( 2nd `  B
) ) ) )
Distinct variable groups:    r, q, x, y, A    B, q,
r, x, y

Proof of Theorem recexprlemloc
Dummy variables  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 7276 . . . . . . . . 9  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
2 prnmaxl 7289 . . . . . . . . 9  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  ( *Q `  r
)  e.  ( 1st `  A ) )  ->  E. u  e.  ( 1st `  A ) ( *Q `  r ) 
<Q  u )
31, 2sylan 281 . . . . . . . 8  |-  ( ( A  e.  P.  /\  ( *Q `  r )  e.  ( 1st `  A
) )  ->  E. u  e.  ( 1st `  A
) ( *Q `  r )  <Q  u
)
43adantlr 468 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  q  <Q  r )  /\  ( *Q `  r
)  e.  ( 1st `  A ) )  ->  E. u  e.  ( 1st `  A ) ( *Q `  r ) 
<Q  u )
5 simprr 521 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
P.  /\  q  <Q  r )  /\  ( *Q
`  r )  e.  ( 1st `  A
) )  /\  (
u  e.  ( 1st `  A )  /\  ( *Q `  r )  <Q  u ) )  -> 
( *Q `  r
)  <Q  u )
6 elprnql 7282 . . . . . . . . . . . . . 14  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  u  e.  ( 1st `  A ) )  ->  u  e.  Q. )
71, 6sylan 281 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  u  e.  ( 1st `  A ) )  ->  u  e.  Q. )
87ad2ant2r 500 . . . . . . . . . . . 12  |-  ( ( ( A  e.  P.  /\  q  <Q  r )  /\  ( u  e.  ( 1st `  A )  /\  ( *Q `  r )  <Q  u
) )  ->  u  e.  Q. )
98adantlr 468 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
P.  /\  q  <Q  r )  /\  ( *Q
`  r )  e.  ( 1st `  A
) )  /\  (
u  e.  ( 1st `  A )  /\  ( *Q `  r )  <Q  u ) )  ->  u  e.  Q. )
10 recrecnq 7195 . . . . . . . . . . 11  |-  ( u  e.  Q.  ->  ( *Q `  ( *Q `  u ) )  =  u )
119, 10syl 14 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
P.  /\  q  <Q  r )  /\  ( *Q
`  r )  e.  ( 1st `  A
) )  /\  (
u  e.  ( 1st `  A )  /\  ( *Q `  r )  <Q  u ) )  -> 
( *Q `  ( *Q `  u ) )  =  u )
125, 11breqtrrd 3951 . . . . . . . . 9  |-  ( ( ( ( A  e. 
P.  /\  q  <Q  r )  /\  ( *Q
`  r )  e.  ( 1st `  A
) )  /\  (
u  e.  ( 1st `  A )  /\  ( *Q `  r )  <Q  u ) )  -> 
( *Q `  r
)  <Q  ( *Q `  ( *Q `  u ) ) )
13 recclnq 7193 . . . . . . . . . . 11  |-  ( u  e.  Q.  ->  ( *Q `  u )  e. 
Q. )
149, 13syl 14 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
P.  /\  q  <Q  r )  /\  ( *Q
`  r )  e.  ( 1st `  A
) )  /\  (
u  e.  ( 1st `  A )  /\  ( *Q `  r )  <Q  u ) )  -> 
( *Q `  u
)  e.  Q. )
15 ltrelnq 7166 . . . . . . . . . . . . . 14  |-  <Q  C_  ( Q.  X.  Q. )
1615brel 4586 . . . . . . . . . . . . 13  |-  ( q 
<Q  r  ->  ( q  e.  Q.  /\  r  e.  Q. ) )
1716adantl 275 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  q  <Q  r )  -> 
( q  e.  Q.  /\  r  e.  Q. )
)
1817ad2antrr 479 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
P.  /\  q  <Q  r )  /\  ( *Q
`  r )  e.  ( 1st `  A
) )  /\  (
u  e.  ( 1st `  A )  /\  ( *Q `  r )  <Q  u ) )  -> 
( q  e.  Q.  /\  r  e.  Q. )
)
1918simprd 113 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
P.  /\  q  <Q  r )  /\  ( *Q
`  r )  e.  ( 1st `  A
) )  /\  (
u  e.  ( 1st `  A )  /\  ( *Q `  r )  <Q  u ) )  -> 
r  e.  Q. )
20 ltrnqg 7221 . . . . . . . . . 10  |-  ( ( ( *Q `  u
)  e.  Q.  /\  r  e.  Q. )  ->  ( ( *Q `  u )  <Q  r  <->  ( *Q `  r ) 
<Q  ( *Q `  ( *Q `  u ) ) ) )
2114, 19, 20syl2anc 408 . . . . . . . . 9  |-  ( ( ( ( A  e. 
P.  /\  q  <Q  r )  /\  ( *Q
`  r )  e.  ( 1st `  A
) )  /\  (
u  e.  ( 1st `  A )  /\  ( *Q `  r )  <Q  u ) )  -> 
( ( *Q `  u )  <Q  r  <->  ( *Q `  r ) 
<Q  ( *Q `  ( *Q `  u ) ) ) )
2212, 21mpbird 166 . . . . . . . 8  |-  ( ( ( ( A  e. 
P.  /\  q  <Q  r )  /\  ( *Q
`  r )  e.  ( 1st `  A
) )  /\  (
u  e.  ( 1st `  A )  /\  ( *Q `  r )  <Q  u ) )  -> 
( *Q `  u
)  <Q  r )
23 simprl 520 . . . . . . . . 9  |-  ( ( ( ( A  e. 
P.  /\  q  <Q  r )  /\  ( *Q
`  r )  e.  ( 1st `  A
) )  /\  (
u  e.  ( 1st `  A )  /\  ( *Q `  r )  <Q  u ) )  ->  u  e.  ( 1st `  A ) )
2411, 23eqeltrd 2214 . . . . . . . 8  |-  ( ( ( ( A  e. 
P.  /\  q  <Q  r )  /\  ( *Q
`  r )  e.  ( 1st `  A
) )  /\  (
u  e.  ( 1st `  A )  /\  ( *Q `  r )  <Q  u ) )  -> 
( *Q `  ( *Q `  u ) )  e.  ( 1st `  A
) )
25 breq1 3927 . . . . . . . . . . . 12  |-  ( y  =  ( *Q `  u )  ->  (
y  <Q  r  <->  ( *Q `  u )  <Q  r
) )
26 fveq2 5414 . . . . . . . . . . . . 13  |-  ( y  =  ( *Q `  u )  ->  ( *Q `  y )  =  ( *Q `  ( *Q `  u ) ) )
2726eleq1d 2206 . . . . . . . . . . . 12  |-  ( y  =  ( *Q `  u )  ->  (
( *Q `  y
)  e.  ( 1st `  A )  <->  ( *Q `  ( *Q `  u
) )  e.  ( 1st `  A ) ) )
2825, 27anbi12d 464 . . . . . . . . . . 11  |-  ( y  =  ( *Q `  u )  ->  (
( y  <Q  r  /\  ( *Q `  y
)  e.  ( 1st `  A ) )  <->  ( ( *Q `  u )  <Q 
r  /\  ( *Q `  ( *Q `  u
) )  e.  ( 1st `  A ) ) ) )
2928spcegv 2769 . . . . . . . . . 10  |-  ( ( *Q `  u )  e.  Q.  ->  (
( ( *Q `  u )  <Q  r  /\  ( *Q `  ( *Q `  u ) )  e.  ( 1st `  A
) )  ->  E. y
( y  <Q  r  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) ) )
30 recexpr.1 . . . . . . . . . . 11  |-  B  = 
<. { x  |  E. y ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  {
x  |  E. y
( y  <Q  x  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) }
>.
3130recexprlemelu 7424 . . . . . . . . . 10  |-  ( r  e.  ( 2nd `  B
)  <->  E. y ( y 
<Q  r  /\  ( *Q `  y )  e.  ( 1st `  A
) ) )
3229, 31syl6ibr 161 . . . . . . . . 9  |-  ( ( *Q `  u )  e.  Q.  ->  (
( ( *Q `  u )  <Q  r  /\  ( *Q `  ( *Q `  u ) )  e.  ( 1st `  A
) )  ->  r  e.  ( 2nd `  B
) ) )
3314, 32syl 14 . . . . . . . 8  |-  ( ( ( ( A  e. 
P.  /\  q  <Q  r )  /\  ( *Q
`  r )  e.  ( 1st `  A
) )  /\  (
u  e.  ( 1st `  A )  /\  ( *Q `  r )  <Q  u ) )  -> 
( ( ( *Q
`  u )  <Q 
r  /\  ( *Q `  ( *Q `  u
) )  e.  ( 1st `  A ) )  ->  r  e.  ( 2nd `  B ) ) )
3422, 24, 33mp2and 429 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  q  <Q  r )  /\  ( *Q
`  r )  e.  ( 1st `  A
) )  /\  (
u  e.  ( 1st `  A )  /\  ( *Q `  r )  <Q  u ) )  -> 
r  e.  ( 2nd `  B ) )
354, 34rexlimddv 2552 . . . . . 6  |-  ( ( ( A  e.  P.  /\  q  <Q  r )  /\  ( *Q `  r
)  e.  ( 1st `  A ) )  -> 
r  e.  ( 2nd `  B ) )
3635olcd 723 . . . . 5  |-  ( ( ( A  e.  P.  /\  q  <Q  r )  /\  ( *Q `  r
)  e.  ( 1st `  A ) )  -> 
( q  e.  ( 1st `  B )  \/  r  e.  ( 2nd `  B ) ) )
37 prnminu 7290 . . . . . . . . 9  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  ( *Q `  q
)  e.  ( 2nd `  A ) )  ->  E. v  e.  ( 2nd `  A ) v 
<Q  ( *Q `  q
) )
381, 37sylan 281 . . . . . . . 8  |-  ( ( A  e.  P.  /\  ( *Q `  q )  e.  ( 2nd `  A
) )  ->  E. v  e.  ( 2nd `  A
) v  <Q  ( *Q `  q ) )
3938adantlr 468 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  q  <Q  r )  /\  ( *Q `  q
)  e.  ( 2nd `  A ) )  ->  E. v  e.  ( 2nd `  A ) v 
<Q  ( *Q `  q
) )
40 elprnqu 7283 . . . . . . . . . . . . . 14  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  v  e.  ( 2nd `  A ) )  -> 
v  e.  Q. )
411, 40sylan 281 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  v  e.  ( 2nd `  A ) )  -> 
v  e.  Q. )
4241adantlr 468 . . . . . . . . . . . 12  |-  ( ( ( A  e.  P.  /\  q  <Q  r )  /\  v  e.  ( 2nd `  A ) )  ->  v  e.  Q. )
4342ad2ant2r 500 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
P.  /\  q  <Q  r )  /\  ( *Q
`  q )  e.  ( 2nd `  A
) )  /\  (
v  e.  ( 2nd `  A )  /\  v  <Q  ( *Q `  q
) ) )  -> 
v  e.  Q. )
44 recrecnq 7195 . . . . . . . . . . 11  |-  ( v  e.  Q.  ->  ( *Q `  ( *Q `  v ) )  =  v )
4543, 44syl 14 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
P.  /\  q  <Q  r )  /\  ( *Q
`  q )  e.  ( 2nd `  A
) )  /\  (
v  e.  ( 2nd `  A )  /\  v  <Q  ( *Q `  q
) ) )  -> 
( *Q `  ( *Q `  v ) )  =  v )
46 simprr 521 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
P.  /\  q  <Q  r )  /\  ( *Q
`  q )  e.  ( 2nd `  A
) )  /\  (
v  e.  ( 2nd `  A )  /\  v  <Q  ( *Q `  q
) ) )  -> 
v  <Q  ( *Q `  q ) )
4745, 46eqbrtrd 3945 . . . . . . . . 9  |-  ( ( ( ( A  e. 
P.  /\  q  <Q  r )  /\  ( *Q
`  q )  e.  ( 2nd `  A
) )  /\  (
v  e.  ( 2nd `  A )  /\  v  <Q  ( *Q `  q
) ) )  -> 
( *Q `  ( *Q `  v ) ) 
<Q  ( *Q `  q
) )
4817ad2antrr 479 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
P.  /\  q  <Q  r )  /\  ( *Q
`  q )  e.  ( 2nd `  A
) )  /\  (
v  e.  ( 2nd `  A )  /\  v  <Q  ( *Q `  q
) ) )  -> 
( q  e.  Q.  /\  r  e.  Q. )
)
4948simpld 111 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
P.  /\  q  <Q  r )  /\  ( *Q
`  q )  e.  ( 2nd `  A
) )  /\  (
v  e.  ( 2nd `  A )  /\  v  <Q  ( *Q `  q
) ) )  -> 
q  e.  Q. )
50 recclnq 7193 . . . . . . . . . . 11  |-  ( v  e.  Q.  ->  ( *Q `  v )  e. 
Q. )
5143, 50syl 14 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
P.  /\  q  <Q  r )  /\  ( *Q
`  q )  e.  ( 2nd `  A
) )  /\  (
v  e.  ( 2nd `  A )  /\  v  <Q  ( *Q `  q
) ) )  -> 
( *Q `  v
)  e.  Q. )
52 ltrnqg 7221 . . . . . . . . . 10  |-  ( ( q  e.  Q.  /\  ( *Q `  v )  e.  Q. )  -> 
( q  <Q  ( *Q `  v )  <->  ( *Q `  ( *Q `  v
) )  <Q  ( *Q `  q ) ) )
5349, 51, 52syl2anc 408 . . . . . . . . 9  |-  ( ( ( ( A  e. 
P.  /\  q  <Q  r )  /\  ( *Q
`  q )  e.  ( 2nd `  A
) )  /\  (
v  e.  ( 2nd `  A )  /\  v  <Q  ( *Q `  q
) ) )  -> 
( q  <Q  ( *Q `  v )  <->  ( *Q `  ( *Q `  v
) )  <Q  ( *Q `  q ) ) )
5447, 53mpbird 166 . . . . . . . 8  |-  ( ( ( ( A  e. 
P.  /\  q  <Q  r )  /\  ( *Q
`  q )  e.  ( 2nd `  A
) )  /\  (
v  e.  ( 2nd `  A )  /\  v  <Q  ( *Q `  q
) ) )  -> 
q  <Q  ( *Q `  v ) )
55 simprl 520 . . . . . . . . 9  |-  ( ( ( ( A  e. 
P.  /\  q  <Q  r )  /\  ( *Q
`  q )  e.  ( 2nd `  A
) )  /\  (
v  e.  ( 2nd `  A )  /\  v  <Q  ( *Q `  q
) ) )  -> 
v  e.  ( 2nd `  A ) )
5645, 55eqeltrd 2214 . . . . . . . 8  |-  ( ( ( ( A  e. 
P.  /\  q  <Q  r )  /\  ( *Q
`  q )  e.  ( 2nd `  A
) )  /\  (
v  e.  ( 2nd `  A )  /\  v  <Q  ( *Q `  q
) ) )  -> 
( *Q `  ( *Q `  v ) )  e.  ( 2nd `  A
) )
57 breq2 3928 . . . . . . . . . . . 12  |-  ( y  =  ( *Q `  v )  ->  (
q  <Q  y  <->  q  <Q  ( *Q `  v ) ) )
58 fveq2 5414 . . . . . . . . . . . . 13  |-  ( y  =  ( *Q `  v )  ->  ( *Q `  y )  =  ( *Q `  ( *Q `  v ) ) )
5958eleq1d 2206 . . . . . . . . . . . 12  |-  ( y  =  ( *Q `  v )  ->  (
( *Q `  y
)  e.  ( 2nd `  A )  <->  ( *Q `  ( *Q `  v
) )  e.  ( 2nd `  A ) ) )
6057, 59anbi12d 464 . . . . . . . . . . 11  |-  ( y  =  ( *Q `  v )  ->  (
( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  <->  ( q  <Q  ( *Q `  v
)  /\  ( *Q `  ( *Q `  v
) )  e.  ( 2nd `  A ) ) ) )
6160spcegv 2769 . . . . . . . . . 10  |-  ( ( *Q `  v )  e.  Q.  ->  (
( q  <Q  ( *Q `  v )  /\  ( *Q `  ( *Q
`  v ) )  e.  ( 2nd `  A
) )  ->  E. y
( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) ) ) )
6230recexprlemell 7423 . . . . . . . . . 10  |-  ( q  e.  ( 1st `  B
)  <->  E. y ( q 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) )
6361, 62syl6ibr 161 . . . . . . . . 9  |-  ( ( *Q `  v )  e.  Q.  ->  (
( q  <Q  ( *Q `  v )  /\  ( *Q `  ( *Q
`  v ) )  e.  ( 2nd `  A
) )  ->  q  e.  ( 1st `  B
) ) )
6451, 63syl 14 . . . . . . . 8  |-  ( ( ( ( A  e. 
P.  /\  q  <Q  r )  /\  ( *Q
`  q )  e.  ( 2nd `  A
) )  /\  (
v  e.  ( 2nd `  A )  /\  v  <Q  ( *Q `  q
) ) )  -> 
( ( q  <Q 
( *Q `  v
)  /\  ( *Q `  ( *Q `  v
) )  e.  ( 2nd `  A ) )  ->  q  e.  ( 1st `  B ) ) )
6554, 56, 64mp2and 429 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  q  <Q  r )  /\  ( *Q
`  q )  e.  ( 2nd `  A
) )  /\  (
v  e.  ( 2nd `  A )  /\  v  <Q  ( *Q `  q
) ) )  -> 
q  e.  ( 1st `  B ) )
6639, 65rexlimddv 2552 . . . . . 6  |-  ( ( ( A  e.  P.  /\  q  <Q  r )  /\  ( *Q `  q
)  e.  ( 2nd `  A ) )  -> 
q  e.  ( 1st `  B ) )
6766orcd 722 . . . . 5  |-  ( ( ( A  e.  P.  /\  q  <Q  r )  /\  ( *Q `  q
)  e.  ( 2nd `  A ) )  -> 
( q  e.  ( 1st `  B )  \/  r  e.  ( 2nd `  B ) ) )
68 ltrnqi 7222 . . . . . 6  |-  ( q 
<Q  r  ->  ( *Q
`  r )  <Q 
( *Q `  q
) )
69 prloc 7292 . . . . . 6  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  ( *Q `  r
)  <Q  ( *Q `  q ) )  -> 
( ( *Q `  r )  e.  ( 1st `  A )  \/  ( *Q `  q )  e.  ( 2nd `  A ) ) )
701, 68, 69syl2an 287 . . . . 5  |-  ( ( A  e.  P.  /\  q  <Q  r )  -> 
( ( *Q `  r )  e.  ( 1st `  A )  \/  ( *Q `  q )  e.  ( 2nd `  A ) ) )
7136, 67, 70mpjaodan 787 . . . 4  |-  ( ( A  e.  P.  /\  q  <Q  r )  -> 
( q  e.  ( 1st `  B )  \/  r  e.  ( 2nd `  B ) ) )
7271ex 114 . . 3  |-  ( A  e.  P.  ->  (
q  <Q  r  ->  (
q  e.  ( 1st `  B )  \/  r  e.  ( 2nd `  B
) ) ) )
7372ralrimivw 2504 . 2  |-  ( A  e.  P.  ->  A. r  e.  Q.  ( q  <Q 
r  ->  ( q  e.  ( 1st `  B
)  \/  r  e.  ( 2nd `  B
) ) ) )
7473ralrimivw 2504 1  |-  ( A  e.  P.  ->  A. q  e.  Q.  A. r  e. 
Q.  ( q  <Q 
r  ->  ( q  e.  ( 1st `  B
)  \/  r  e.  ( 2nd `  B
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697    = wceq 1331   E.wex 1468    e. wcel 1480   {cab 2123   A.wral 2414   E.wrex 2415   <.cop 3525   class class class wbr 3924   ` cfv 5118   1stc1st 6029   2ndc2nd 6030   Q.cnq 7081   *Qcrq 7085    <Q cltq 7086   P.cnp 7092
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-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-mi 7107  df-lti 7108  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-inp 7267
This theorem is referenced by:  recexprlempr  7433
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