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Theorem recexprlemloc 6787
Description:  B is located. Lemma for recexpr 6794. (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 6631 . . . . . . . . 9  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
2 prnmaxl 6644 . . . . . . . . 9  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  ( *Q `  r
)  e.  ( 1st `  A ) )  ->  E. u  e.  ( 1st `  A ) ( *Q `  r ) 
<Q  u )
31, 2sylan 271 . . . . . . . 8  |-  ( ( A  e.  P.  /\  ( *Q `  r )  e.  ( 1st `  A
) )  ->  E. u  e.  ( 1st `  A
) ( *Q `  r )  <Q  u
)
43adantlr 454 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  q  <Q  r )  /\  ( *Q `  r
)  e.  ( 1st `  A ) )  ->  E. u  e.  ( 1st `  A ) ( *Q `  r ) 
<Q  u )
5 simprr 492 . . . . . . . . . 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 6637 . . . . . . . . . . . . . 14  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  u  e.  ( 1st `  A ) )  ->  u  e.  Q. )
71, 6sylan 271 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  u  e.  ( 1st `  A ) )  ->  u  e.  Q. )
87ad2ant2r 486 . . . . . . . . . . . 12  |-  ( ( ( A  e.  P.  /\  q  <Q  r )  /\  ( u  e.  ( 1st `  A )  /\  ( *Q `  r )  <Q  u
) )  ->  u  e.  Q. )
98adantlr 454 . . . . . . . . . . 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 6550 . . . . . . . . . . 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 3818 . . . . . . . . 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 6548 . . . . . . . . . . 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 6521 . . . . . . . . . . . . . 14  |-  <Q  C_  ( Q.  X.  Q. )
1615brel 4420 . . . . . . . . . . . . 13  |-  ( q 
<Q  r  ->  ( q  e.  Q.  /\  r  e.  Q. ) )
1716adantl 266 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  q  <Q  r )  -> 
( q  e.  Q.  /\  r  e.  Q. )
)
1817ad2antrr 465 . . . . . . . . . . 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 111 . . . . . . . . . 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 6576 . . . . . . . . . 10  |-  ( ( ( *Q `  u
)  e.  Q.  /\  r  e.  Q. )  ->  ( ( *Q `  u )  <Q  r  <->  ( *Q `  r ) 
<Q  ( *Q `  ( *Q `  u ) ) ) )
2114, 19, 20syl2anc 397 . . . . . . . . 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 160 . . . . . . . 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 491 . . . . . . . . 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 2130 . . . . . . . 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 3795 . . . . . . . . . . . 12  |-  ( y  =  ( *Q `  u )  ->  (
y  <Q  r  <->  ( *Q `  u )  <Q  r
) )
26 fveq2 5206 . . . . . . . . . . . . 13  |-  ( y  =  ( *Q `  u )  ->  ( *Q `  y )  =  ( *Q `  ( *Q `  u ) ) )
2726eleq1d 2122 . . . . . . . . . . . 12  |-  ( y  =  ( *Q `  u )  ->  (
( *Q `  y
)  e.  ( 1st `  A )  <->  ( *Q `  ( *Q `  u
) )  e.  ( 1st `  A ) ) )
2825, 27anbi12d 450 . . . . . . . . . . 11  |-  ( y  =  ( *Q `  u )  ->  (
( y  <Q  r  /\  ( *Q `  y
)  e.  ( 1st `  A ) )  <->  ( ( *Q `  u )  <Q 
r  /\  ( *Q `  ( *Q `  u
) )  e.  ( 1st `  A ) ) ) )
2928spcegv 2658 . . . . . . . . . 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 6779 . . . . . . . . . 10  |-  ( r  e.  ( 2nd `  B
)  <->  E. y ( y 
<Q  r  /\  ( *Q `  y )  e.  ( 1st `  A
) ) )
3229, 31syl6ibr 155 . . . . . . . . 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 417 . . . . . . 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 2454 . . . . . 6  |-  ( ( ( A  e.  P.  /\  q  <Q  r )  /\  ( *Q `  r
)  e.  ( 1st `  A ) )  -> 
r  e.  ( 2nd `  B ) )
3635olcd 663 . . . . 5  |-  ( ( ( A  e.  P.  /\  q  <Q  r )  /\  ( *Q `  r
)  e.  ( 1st `  A ) )  -> 
( q  e.  ( 1st `  B )  \/  r  e.  ( 2nd `  B ) ) )
37 prnminu 6645 . . . . . . . . 9  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  ( *Q `  q
)  e.  ( 2nd `  A ) )  ->  E. v  e.  ( 2nd `  A ) v 
<Q  ( *Q `  q
) )
381, 37sylan 271 . . . . . . . 8  |-  ( ( A  e.  P.  /\  ( *Q `  q )  e.  ( 2nd `  A
) )  ->  E. v  e.  ( 2nd `  A
) v  <Q  ( *Q `  q ) )
3938adantlr 454 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  q  <Q  r )  /\  ( *Q `  q
)  e.  ( 2nd `  A ) )  ->  E. v  e.  ( 2nd `  A ) v 
<Q  ( *Q `  q
) )
40 elprnqu 6638 . . . . . . . . . . . . . 14  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  v  e.  ( 2nd `  A ) )  -> 
v  e.  Q. )
411, 40sylan 271 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  v  e.  ( 2nd `  A ) )  -> 
v  e.  Q. )
4241adantlr 454 . . . . . . . . . . . 12  |-  ( ( ( A  e.  P.  /\  q  <Q  r )  /\  v  e.  ( 2nd `  A ) )  ->  v  e.  Q. )
4342ad2ant2r 486 . . . . . . . . . . 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 6550 . . . . . . . . . . 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 492 . . . . . . . . . 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 3812 . . . . . . . . 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 465 . . . . . . . . . . 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 109 . . . . . . . . . 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 6548 . . . . . . . . . . 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 6576 . . . . . . . . . 10  |-  ( ( q  e.  Q.  /\  ( *Q `  v )  e.  Q. )  -> 
( q  <Q  ( *Q `  v )  <->  ( *Q `  ( *Q `  v
) )  <Q  ( *Q `  q ) ) )
5349, 51, 52syl2anc 397 . . . . . . . . 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 160 . . . . . . . 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 491 . . . . . . . . 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 2130 . . . . . . . 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 3796 . . . . . . . . . . . 12  |-  ( y  =  ( *Q `  v )  ->  (
q  <Q  y  <->  q  <Q  ( *Q `  v ) ) )
58 fveq2 5206 . . . . . . . . . . . . 13  |-  ( y  =  ( *Q `  v )  ->  ( *Q `  y )  =  ( *Q `  ( *Q `  v ) ) )
5958eleq1d 2122 . . . . . . . . . . . 12  |-  ( y  =  ( *Q `  v )  ->  (
( *Q `  y
)  e.  ( 2nd `  A )  <->  ( *Q `  ( *Q `  v
) )  e.  ( 2nd `  A ) ) )
6057, 59anbi12d 450 . . . . . . . . . . 11  |-  ( y  =  ( *Q `  v )  ->  (
( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  <->  ( q  <Q  ( *Q `  v
)  /\  ( *Q `  ( *Q `  v
) )  e.  ( 2nd `  A ) ) ) )
6160spcegv 2658 . . . . . . . . . 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 6778 . . . . . . . . . 10  |-  ( q  e.  ( 1st `  B
)  <->  E. y ( q 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) )
6361, 62syl6ibr 155 . . . . . . . . 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 417 . . . . . . 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 2454 . . . . . 6  |-  ( ( ( A  e.  P.  /\  q  <Q  r )  /\  ( *Q `  q
)  e.  ( 2nd `  A ) )  -> 
q  e.  ( 1st `  B ) )
6766orcd 662 . . . . 5  |-  ( ( ( A  e.  P.  /\  q  <Q  r )  /\  ( *Q `  q
)  e.  ( 2nd `  A ) )  -> 
( q  e.  ( 1st `  B )  \/  r  e.  ( 2nd `  B ) ) )
68 ltrnqi 6577 . . . . . 6  |-  ( q 
<Q  r  ->  ( *Q
`  r )  <Q 
( *Q `  q
) )
69 prloc 6647 . . . . . 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 277 . . . . 5  |-  ( ( A  e.  P.  /\  q  <Q  r )  -> 
( ( *Q `  r )  e.  ( 1st `  A )  \/  ( *Q `  q )  e.  ( 2nd `  A ) ) )
7136, 67, 70mpjaodan 722 . . . 4  |-  ( ( A  e.  P.  /\  q  <Q  r )  -> 
( q  e.  ( 1st `  B )  \/  r  e.  ( 2nd `  B ) ) )
7271ex 112 . . 3  |-  ( A  e.  P.  ->  (
q  <Q  r  ->  (
q  e.  ( 1st `  B )  \/  r  e.  ( 2nd `  B
) ) ) )
7372ralrimivw 2410 . 2  |-  ( A  e.  P.  ->  A. r  e.  Q.  ( q  <Q 
r  ->  ( q  e.  ( 1st `  B
)  \/  r  e.  ( 2nd `  B
) ) ) )
7473ralrimivw 2410 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 101    <-> wb 102    \/ wo 639    = wceq 1259   E.wex 1397    e. wcel 1409   {cab 2042   A.wral 2323   E.wrex 2324   <.cop 3406   class class class wbr 3792   ` cfv 4930   1stc1st 5793   2ndc2nd 5794   Q.cnq 6436   *Qcrq 6440    <Q cltq 6441   P.cnp 6447
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-mi 6462  df-lti 6463  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-inp 6622
This theorem is referenced by:  recexprlempr  6788
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