ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  recexprlemloc Unicode version

Theorem recexprlemloc 7188
Description:  B is located. Lemma for recexpr 7195. (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 7032 . . . . . . . . 9  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
2 prnmaxl 7045 . . . . . . . . 9  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  ( *Q `  r
)  e.  ( 1st `  A ) )  ->  E. u  e.  ( 1st `  A ) ( *Q `  r ) 
<Q  u )
31, 2sylan 277 . . . . . . . 8  |-  ( ( A  e.  P.  /\  ( *Q `  r )  e.  ( 1st `  A
) )  ->  E. u  e.  ( 1st `  A
) ( *Q `  r )  <Q  u
)
43adantlr 461 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  q  <Q  r )  /\  ( *Q `  r
)  e.  ( 1st `  A ) )  ->  E. u  e.  ( 1st `  A ) ( *Q `  r ) 
<Q  u )
5 simprr 499 . . . . . . . . . 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 7038 . . . . . . . . . . . . . 14  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  u  e.  ( 1st `  A ) )  ->  u  e.  Q. )
71, 6sylan 277 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  u  e.  ( 1st `  A ) )  ->  u  e.  Q. )
87ad2ant2r 493 . . . . . . . . . . . 12  |-  ( ( ( A  e.  P.  /\  q  <Q  r )  /\  ( u  e.  ( 1st `  A )  /\  ( *Q `  r )  <Q  u
) )  ->  u  e.  Q. )
98adantlr 461 . . . . . . . . . . 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 6951 . . . . . . . . . . 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 3871 . . . . . . . . 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 6949 . . . . . . . . . . 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 6922 . . . . . . . . . . . . . 14  |-  <Q  C_  ( Q.  X.  Q. )
1615brel 4490 . . . . . . . . . . . . 13  |-  ( q 
<Q  r  ->  ( q  e.  Q.  /\  r  e.  Q. ) )
1716adantl 271 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  q  <Q  r )  -> 
( q  e.  Q.  /\  r  e.  Q. )
)
1817ad2antrr 472 . . . . . . . . . . 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 112 . . . . . . . . . 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 6977 . . . . . . . . . 10  |-  ( ( ( *Q `  u
)  e.  Q.  /\  r  e.  Q. )  ->  ( ( *Q `  u )  <Q  r  <->  ( *Q `  r ) 
<Q  ( *Q `  ( *Q `  u ) ) ) )
2114, 19, 20syl2anc 403 . . . . . . . . 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 165 . . . . . . . 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 498 . . . . . . . . 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 2164 . . . . . . . 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 3848 . . . . . . . . . . . 12  |-  ( y  =  ( *Q `  u )  ->  (
y  <Q  r  <->  ( *Q `  u )  <Q  r
) )
26 fveq2 5305 . . . . . . . . . . . . 13  |-  ( y  =  ( *Q `  u )  ->  ( *Q `  y )  =  ( *Q `  ( *Q `  u ) ) )
2726eleq1d 2156 . . . . . . . . . . . 12  |-  ( y  =  ( *Q `  u )  ->  (
( *Q `  y
)  e.  ( 1st `  A )  <->  ( *Q `  ( *Q `  u
) )  e.  ( 1st `  A ) ) )
2825, 27anbi12d 457 . . . . . . . . . . 11  |-  ( y  =  ( *Q `  u )  ->  (
( y  <Q  r  /\  ( *Q `  y
)  e.  ( 1st `  A ) )  <->  ( ( *Q `  u )  <Q 
r  /\  ( *Q `  ( *Q `  u
) )  e.  ( 1st `  A ) ) ) )
2928spcegv 2707 . . . . . . . . . 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 7180 . . . . . . . . . 10  |-  ( r  e.  ( 2nd `  B
)  <->  E. y ( y 
<Q  r  /\  ( *Q `  y )  e.  ( 1st `  A
) ) )
3229, 31syl6ibr 160 . . . . . . . . 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 424 . . . . . . 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 2493 . . . . . 6  |-  ( ( ( A  e.  P.  /\  q  <Q  r )  /\  ( *Q `  r
)  e.  ( 1st `  A ) )  -> 
r  e.  ( 2nd `  B ) )
3635olcd 688 . . . . 5  |-  ( ( ( A  e.  P.  /\  q  <Q  r )  /\  ( *Q `  r
)  e.  ( 1st `  A ) )  -> 
( q  e.  ( 1st `  B )  \/  r  e.  ( 2nd `  B ) ) )
37 prnminu 7046 . . . . . . . . 9  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  ( *Q `  q
)  e.  ( 2nd `  A ) )  ->  E. v  e.  ( 2nd `  A ) v 
<Q  ( *Q `  q
) )
381, 37sylan 277 . . . . . . . 8  |-  ( ( A  e.  P.  /\  ( *Q `  q )  e.  ( 2nd `  A
) )  ->  E. v  e.  ( 2nd `  A
) v  <Q  ( *Q `  q ) )
3938adantlr 461 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  q  <Q  r )  /\  ( *Q `  q
)  e.  ( 2nd `  A ) )  ->  E. v  e.  ( 2nd `  A ) v 
<Q  ( *Q `  q
) )
40 elprnqu 7039 . . . . . . . . . . . . . 14  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  v  e.  ( 2nd `  A ) )  -> 
v  e.  Q. )
411, 40sylan 277 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  v  e.  ( 2nd `  A ) )  -> 
v  e.  Q. )
4241adantlr 461 . . . . . . . . . . . 12  |-  ( ( ( A  e.  P.  /\  q  <Q  r )  /\  v  e.  ( 2nd `  A ) )  ->  v  e.  Q. )
4342ad2ant2r 493 . . . . . . . . . . 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 6951 . . . . . . . . . . 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 499 . . . . . . . . . 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 3865 . . . . . . . . 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 472 . . . . . . . . . . 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 110 . . . . . . . . . 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 6949 . . . . . . . . . . 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 6977 . . . . . . . . . 10  |-  ( ( q  e.  Q.  /\  ( *Q `  v )  e.  Q. )  -> 
( q  <Q  ( *Q `  v )  <->  ( *Q `  ( *Q `  v
) )  <Q  ( *Q `  q ) ) )
5349, 51, 52syl2anc 403 . . . . . . . . 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 165 . . . . . . . 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 498 . . . . . . . . 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 2164 . . . . . . . 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 3849 . . . . . . . . . . . 12  |-  ( y  =  ( *Q `  v )  ->  (
q  <Q  y  <->  q  <Q  ( *Q `  v ) ) )
58 fveq2 5305 . . . . . . . . . . . . 13  |-  ( y  =  ( *Q `  v )  ->  ( *Q `  y )  =  ( *Q `  ( *Q `  v ) ) )
5958eleq1d 2156 . . . . . . . . . . . 12  |-  ( y  =  ( *Q `  v )  ->  (
( *Q `  y
)  e.  ( 2nd `  A )  <->  ( *Q `  ( *Q `  v
) )  e.  ( 2nd `  A ) ) )
6057, 59anbi12d 457 . . . . . . . . . . 11  |-  ( y  =  ( *Q `  v )  ->  (
( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  <->  ( q  <Q  ( *Q `  v
)  /\  ( *Q `  ( *Q `  v
) )  e.  ( 2nd `  A ) ) ) )
6160spcegv 2707 . . . . . . . . . 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 7179 . . . . . . . . . 10  |-  ( q  e.  ( 1st `  B
)  <->  E. y ( q 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) )
6361, 62syl6ibr 160 . . . . . . . . 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 424 . . . . . . 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 2493 . . . . . 6  |-  ( ( ( A  e.  P.  /\  q  <Q  r )  /\  ( *Q `  q
)  e.  ( 2nd `  A ) )  -> 
q  e.  ( 1st `  B ) )
6766orcd 687 . . . . 5  |-  ( ( ( A  e.  P.  /\  q  <Q  r )  /\  ( *Q `  q
)  e.  ( 2nd `  A ) )  -> 
( q  e.  ( 1st `  B )  \/  r  e.  ( 2nd `  B ) ) )
68 ltrnqi 6978 . . . . . 6  |-  ( q 
<Q  r  ->  ( *Q
`  r )  <Q 
( *Q `  q
) )
69 prloc 7048 . . . . . 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 283 . . . . 5  |-  ( ( A  e.  P.  /\  q  <Q  r )  -> 
( ( *Q `  r )  e.  ( 1st `  A )  \/  ( *Q `  q )  e.  ( 2nd `  A ) ) )
7136, 67, 70mpjaodan 747 . . . 4  |-  ( ( A  e.  P.  /\  q  <Q  r )  -> 
( q  e.  ( 1st `  B )  \/  r  e.  ( 2nd `  B ) ) )
7271ex 113 . . 3  |-  ( A  e.  P.  ->  (
q  <Q  r  ->  (
q  e.  ( 1st `  B )  \/  r  e.  ( 2nd `  B
) ) ) )
7372ralrimivw 2447 . 2  |-  ( A  e.  P.  ->  A. r  e.  Q.  ( q  <Q 
r  ->  ( q  e.  ( 1st `  B
)  \/  r  e.  ( 2nd `  B
) ) ) )
7473ralrimivw 2447 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 102    <-> wb 103    \/ wo 664    = wceq 1289   E.wex 1426    e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360   <.cop 3449   class class class wbr 3845   ` cfv 5015   1stc1st 5909   2ndc2nd 5910   Q.cnq 6837   *Qcrq 6841    <Q cltq 6842   P.cnp 6848
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-eprel 4116  df-id 4120  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-1o 6181  df-oadd 6185  df-omul 6186  df-er 6290  df-ec 6292  df-qs 6296  df-ni 6861  df-mi 6863  df-lti 6864  df-mpq 6902  df-enq 6904  df-nqqs 6905  df-mqqs 6907  df-1nqqs 6908  df-rq 6909  df-ltnqqs 6910  df-inp 7023
This theorem is referenced by:  recexprlempr  7189
  Copyright terms: Public domain W3C validator