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

Theorem recexprlemloc 7632
Description:  B is located. Lemma for recexpr 7639. (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 7476 . . . . . . . . 9  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
2 prnmaxl 7489 . . . . . . . . 9  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  ( *Q `  r
)  e.  ( 1st `  A ) )  ->  E. u  e.  ( 1st `  A ) ( *Q `  r ) 
<Q  u )
31, 2sylan 283 . . . . . . . 8  |-  ( ( A  e.  P.  /\  ( *Q `  r )  e.  ( 1st `  A
) )  ->  E. u  e.  ( 1st `  A
) ( *Q `  r )  <Q  u
)
43adantlr 477 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  q  <Q  r )  /\  ( *Q `  r
)  e.  ( 1st `  A ) )  ->  E. u  e.  ( 1st `  A ) ( *Q `  r ) 
<Q  u )
5 simprr 531 . . . . . . . . . 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 7482 . . . . . . . . . . . . . 14  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  u  e.  ( 1st `  A ) )  ->  u  e.  Q. )
71, 6sylan 283 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  u  e.  ( 1st `  A ) )  ->  u  e.  Q. )
87ad2ant2r 509 . . . . . . . . . . . 12  |-  ( ( ( A  e.  P.  /\  q  <Q  r )  /\  ( u  e.  ( 1st `  A )  /\  ( *Q `  r )  <Q  u
) )  ->  u  e.  Q. )
98adantlr 477 . . . . . . . . . . 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 7395 . . . . . . . . . . 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 4033 . . . . . . . . 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 7393 . . . . . . . . . . 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 7366 . . . . . . . . . . . . . 14  |-  <Q  C_  ( Q.  X.  Q. )
1615brel 4680 . . . . . . . . . . . . 13  |-  ( q 
<Q  r  ->  ( q  e.  Q.  /\  r  e.  Q. ) )
1716adantl 277 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  q  <Q  r )  -> 
( q  e.  Q.  /\  r  e.  Q. )
)
1817ad2antrr 488 . . . . . . . . . . 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 114 . . . . . . . . . 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 7421 . . . . . . . . . 10  |-  ( ( ( *Q `  u
)  e.  Q.  /\  r  e.  Q. )  ->  ( ( *Q `  u )  <Q  r  <->  ( *Q `  r ) 
<Q  ( *Q `  ( *Q `  u ) ) ) )
2114, 19, 20syl2anc 411 . . . . . . . . 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 167 . . . . . . . 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 529 . . . . . . . . 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 2254 . . . . . . . 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 4008 . . . . . . . . . . . 12  |-  ( y  =  ( *Q `  u )  ->  (
y  <Q  r  <->  ( *Q `  u )  <Q  r
) )
26 fveq2 5517 . . . . . . . . . . . . 13  |-  ( y  =  ( *Q `  u )  ->  ( *Q `  y )  =  ( *Q `  ( *Q `  u ) ) )
2726eleq1d 2246 . . . . . . . . . . . 12  |-  ( y  =  ( *Q `  u )  ->  (
( *Q `  y
)  e.  ( 1st `  A )  <->  ( *Q `  ( *Q `  u
) )  e.  ( 1st `  A ) ) )
2825, 27anbi12d 473 . . . . . . . . . . 11  |-  ( y  =  ( *Q `  u )  ->  (
( y  <Q  r  /\  ( *Q `  y
)  e.  ( 1st `  A ) )  <->  ( ( *Q `  u )  <Q 
r  /\  ( *Q `  ( *Q `  u
) )  e.  ( 1st `  A ) ) ) )
2928spcegv 2827 . . . . . . . . . 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 7624 . . . . . . . . . 10  |-  ( r  e.  ( 2nd `  B
)  <->  E. y ( y 
<Q  r  /\  ( *Q `  y )  e.  ( 1st `  A
) ) )
3229, 31imbitrrdi 162 . . . . . . . . 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 433 . . . . . . 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 2599 . . . . . 6  |-  ( ( ( A  e.  P.  /\  q  <Q  r )  /\  ( *Q `  r
)  e.  ( 1st `  A ) )  -> 
r  e.  ( 2nd `  B ) )
3635olcd 734 . . . . 5  |-  ( ( ( A  e.  P.  /\  q  <Q  r )  /\  ( *Q `  r
)  e.  ( 1st `  A ) )  -> 
( q  e.  ( 1st `  B )  \/  r  e.  ( 2nd `  B ) ) )
37 prnminu 7490 . . . . . . . . 9  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  ( *Q `  q
)  e.  ( 2nd `  A ) )  ->  E. v  e.  ( 2nd `  A ) v 
<Q  ( *Q `  q
) )
381, 37sylan 283 . . . . . . . 8  |-  ( ( A  e.  P.  /\  ( *Q `  q )  e.  ( 2nd `  A
) )  ->  E. v  e.  ( 2nd `  A
) v  <Q  ( *Q `  q ) )
3938adantlr 477 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  q  <Q  r )  /\  ( *Q `  q
)  e.  ( 2nd `  A ) )  ->  E. v  e.  ( 2nd `  A ) v 
<Q  ( *Q `  q
) )
40 elprnqu 7483 . . . . . . . . . . . . . 14  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  v  e.  ( 2nd `  A ) )  -> 
v  e.  Q. )
411, 40sylan 283 . . . . . . . . . . . . 13  |-  ( ( A  e.  P.  /\  v  e.  ( 2nd `  A ) )  -> 
v  e.  Q. )
4241adantlr 477 . . . . . . . . . . . 12  |-  ( ( ( A  e.  P.  /\  q  <Q  r )  /\  v  e.  ( 2nd `  A ) )  ->  v  e.  Q. )
4342ad2ant2r 509 . . . . . . . . . . 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 7395 . . . . . . . . . . 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 531 . . . . . . . . . 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 4027 . . . . . . . . 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 488 . . . . . . . . . . 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 112 . . . . . . . . . 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 7393 . . . . . . . . . . 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 7421 . . . . . . . . . 10  |-  ( ( q  e.  Q.  /\  ( *Q `  v )  e.  Q. )  -> 
( q  <Q  ( *Q `  v )  <->  ( *Q `  ( *Q `  v
) )  <Q  ( *Q `  q ) ) )
5349, 51, 52syl2anc 411 . . . . . . . . 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 167 . . . . . . . 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 529 . . . . . . . . 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 2254 . . . . . . . 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 4009 . . . . . . . . . . . 12  |-  ( y  =  ( *Q `  v )  ->  (
q  <Q  y  <->  q  <Q  ( *Q `  v ) ) )
58 fveq2 5517 . . . . . . . . . . . . 13  |-  ( y  =  ( *Q `  v )  ->  ( *Q `  y )  =  ( *Q `  ( *Q `  v ) ) )
5958eleq1d 2246 . . . . . . . . . . . 12  |-  ( y  =  ( *Q `  v )  ->  (
( *Q `  y
)  e.  ( 2nd `  A )  <->  ( *Q `  ( *Q `  v
) )  e.  ( 2nd `  A ) ) )
6057, 59anbi12d 473 . . . . . . . . . . 11  |-  ( y  =  ( *Q `  v )  ->  (
( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  <->  ( q  <Q  ( *Q `  v
)  /\  ( *Q `  ( *Q `  v
) )  e.  ( 2nd `  A ) ) ) )
6160spcegv 2827 . . . . . . . . . 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 7623 . . . . . . . . . 10  |-  ( q  e.  ( 1st `  B
)  <->  E. y ( q 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) )
6361, 62imbitrrdi 162 . . . . . . . . 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 433 . . . . . . 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 2599 . . . . . 6  |-  ( ( ( A  e.  P.  /\  q  <Q  r )  /\  ( *Q `  q
)  e.  ( 2nd `  A ) )  -> 
q  e.  ( 1st `  B ) )
6766orcd 733 . . . . 5  |-  ( ( ( A  e.  P.  /\  q  <Q  r )  /\  ( *Q `  q
)  e.  ( 2nd `  A ) )  -> 
( q  e.  ( 1st `  B )  \/  r  e.  ( 2nd `  B ) ) )
68 ltrnqi 7422 . . . . . 6  |-  ( q 
<Q  r  ->  ( *Q
`  r )  <Q 
( *Q `  q
) )
69 prloc 7492 . . . . . 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 289 . . . . 5  |-  ( ( A  e.  P.  /\  q  <Q  r )  -> 
( ( *Q `  r )  e.  ( 1st `  A )  \/  ( *Q `  q )  e.  ( 2nd `  A ) ) )
7136, 67, 70mpjaodan 798 . . . 4  |-  ( ( A  e.  P.  /\  q  <Q  r )  -> 
( q  e.  ( 1st `  B )  \/  r  e.  ( 2nd `  B ) ) )
7271ex 115 . . 3  |-  ( A  e.  P.  ->  (
q  <Q  r  ->  (
q  e.  ( 1st `  B )  \/  r  e.  ( 2nd `  B
) ) ) )
7372ralrimivw 2551 . 2  |-  ( A  e.  P.  ->  A. r  e.  Q.  ( q  <Q 
r  ->  ( q  e.  ( 1st `  B
)  \/  r  e.  ( 2nd `  B
) ) ) )
7473ralrimivw 2551 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 104    <-> wb 105    \/ wo 708    = wceq 1353   E.wex 1492    e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   <.cop 3597   class class class wbr 4005   ` cfv 5218   1stc1st 6141   2ndc2nd 6142   Q.cnq 7281   *Qcrq 7285    <Q cltq 7286   P.cnp 7292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-mi 7307  df-lti 7308  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354  df-inp 7467
This theorem is referenced by:  recexprlempr  7633
  Copyright terms: Public domain W3C validator