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Theorem recmulnq 9345
Description: Relationship between reciprocal and multiplication on positive fractions. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
Assertion
Ref Expression
recmulnq  |-  ( A  e.  Q.  ->  (
( *Q `  A
)  =  B  <->  ( A  .Q  B )  =  1Q ) )

Proof of Theorem recmulnq
Dummy variables  x  y  s  r  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 5866 . . . 4  |-  ( *Q
`  A )  e. 
_V
21a1i 11 . . 3  |-  ( A  e.  Q.  ->  ( *Q `  A )  e. 
_V )
3 eleq1 2515 . . 3  |-  ( ( *Q `  A )  =  B  ->  (
( *Q `  A
)  e.  _V  <->  B  e.  _V ) )
42, 3syl5ibcom 220 . 2  |-  ( A  e.  Q.  ->  (
( *Q `  A
)  =  B  ->  B  e.  _V )
)
5 id 22 . . . . . . 7  |-  ( ( A  .Q  B )  =  1Q  ->  ( A  .Q  B )  =  1Q )
6 1nq 9309 . . . . . . 7  |-  1Q  e.  Q.
75, 6syl6eqel 2539 . . . . . 6  |-  ( ( A  .Q  B )  =  1Q  ->  ( A  .Q  B )  e. 
Q. )
8 mulnqf 9330 . . . . . . . 8  |-  .Q  :
( Q.  X.  Q. )
--> Q.
98fdmi 5726 . . . . . . 7  |-  dom  .Q  =  ( Q.  X.  Q. )
10 0nnq 9305 . . . . . . 7  |-  -.  (/)  e.  Q.
119, 10ndmovrcl 6446 . . . . . 6  |-  ( ( A  .Q  B )  e.  Q.  ->  ( A  e.  Q.  /\  B  e.  Q. ) )
127, 11syl 16 . . . . 5  |-  ( ( A  .Q  B )  =  1Q  ->  ( A  e.  Q.  /\  B  e.  Q. ) )
1312simprd 463 . . . 4  |-  ( ( A  .Q  B )  =  1Q  ->  B  e.  Q. )
14 elex 3104 . . . 4  |-  ( B  e.  Q.  ->  B  e.  _V )
1513, 14syl 16 . . 3  |-  ( ( A  .Q  B )  =  1Q  ->  B  e.  _V )
1615a1i 11 . 2  |-  ( A  e.  Q.  ->  (
( A  .Q  B
)  =  1Q  ->  B  e.  _V ) )
17 oveq1 6288 . . . . 5  |-  ( x  =  A  ->  (
x  .Q  y )  =  ( A  .Q  y ) )
1817eqeq1d 2445 . . . 4  |-  ( x  =  A  ->  (
( x  .Q  y
)  =  1Q  <->  ( A  .Q  y )  =  1Q ) )
19 oveq2 6289 . . . . 5  |-  ( y  =  B  ->  ( A  .Q  y )  =  ( A  .Q  B
) )
2019eqeq1d 2445 . . . 4  |-  ( y  =  B  ->  (
( A  .Q  y
)  =  1Q  <->  ( A  .Q  B )  =  1Q ) )
21 nqerid 9314 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  ( /Q `  x )  =  x )
22 relxp 5100 . . . . . . . . . . . 12  |-  Rel  ( N.  X.  N. )
23 elpqn 9306 . . . . . . . . . . . 12  |-  ( x  e.  Q.  ->  x  e.  ( N.  X.  N. ) )
24 1st2nd 6831 . . . . . . . . . . . 12  |-  ( ( Rel  ( N.  X.  N. )  /\  x  e.  ( N.  X.  N. ) )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
2522, 23, 24sylancr 663 . . . . . . . . . . 11  |-  ( x  e.  Q.  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
2625fveq2d 5860 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  ( /Q `  x )  =  ( /Q `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. ) )
2721, 26eqtr3d 2486 . . . . . . . . 9  |-  ( x  e.  Q.  ->  x  =  ( /Q `  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
)
2827oveq1d 6296 . . . . . . . 8  |-  ( x  e.  Q.  ->  (
x  .Q  ( /Q
`  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
)  =  ( ( /Q `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )  .Q  ( /Q `  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
) )
29 mulerpq 9338 . . . . . . . 8  |-  ( ( /Q `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )  .Q  ( /Q `  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
)  =  ( /Q
`  ( <. ( 1st `  x ) ,  ( 2nd `  x
) >.  .pQ  <. ( 2nd `  x ) ,  ( 1st `  x )
>. ) )
3028, 29syl6eq 2500 . . . . . . 7  |-  ( x  e.  Q.  ->  (
x  .Q  ( /Q
`  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
)  =  ( /Q
`  ( <. ( 1st `  x ) ,  ( 2nd `  x
) >.  .pQ  <. ( 2nd `  x ) ,  ( 1st `  x )
>. ) ) )
31 xp1st 6815 . . . . . . . . . . 11  |-  ( x  e.  ( N.  X.  N. )  ->  ( 1st `  x )  e.  N. )
3223, 31syl 16 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  ( 1st `  x )  e. 
N. )
33 xp2nd 6816 . . . . . . . . . . 11  |-  ( x  e.  ( N.  X.  N. )  ->  ( 2nd `  x )  e.  N. )
3423, 33syl 16 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  ( 2nd `  x )  e. 
N. )
35 mulpipq 9321 . . . . . . . . . 10  |-  ( ( ( ( 1st `  x
)  e.  N.  /\  ( 2nd `  x )  e.  N. )  /\  ( ( 2nd `  x
)  e.  N.  /\  ( 1st `  x )  e.  N. ) )  ->  ( <. ( 1st `  x ) ,  ( 2nd `  x
) >.  .pQ  <. ( 2nd `  x ) ,  ( 1st `  x )
>. )  =  <. ( ( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 2nd `  x
)  .N  ( 1st `  x ) ) >.
)
3632, 34, 34, 32, 35syl22anc 1230 . . . . . . . . 9  |-  ( x  e.  Q.  ->  ( <. ( 1st `  x
) ,  ( 2nd `  x ) >.  .pQ  <. ( 2nd `  x ) ,  ( 1st `  x
) >. )  =  <. ( ( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 2nd `  x
)  .N  ( 1st `  x ) ) >.
)
37 mulcompi 9277 . . . . . . . . . 10  |-  ( ( 2nd `  x )  .N  ( 1st `  x
) )  =  ( ( 1st `  x
)  .N  ( 2nd `  x ) )
3837opeq2i 4206 . . . . . . . . 9  |-  <. (
( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 2nd `  x
)  .N  ( 1st `  x ) ) >.  =  <. ( ( 1st `  x )  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x )  .N  ( 2nd `  x ) )
>.
3936, 38syl6eq 2500 . . . . . . . 8  |-  ( x  e.  Q.  ->  ( <. ( 1st `  x
) ,  ( 2nd `  x ) >.  .pQ  <. ( 2nd `  x ) ,  ( 1st `  x
) >. )  =  <. ( ( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x
)  .N  ( 2nd `  x ) ) >.
)
4039fveq2d 5860 . . . . . . 7  |-  ( x  e.  Q.  ->  ( /Q `  ( <. ( 1st `  x ) ,  ( 2nd `  x
) >.  .pQ  <. ( 2nd `  x ) ,  ( 1st `  x )
>. ) )  =  ( /Q `  <. (
( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x
)  .N  ( 2nd `  x ) ) >.
) )
41 nqerid 9314 . . . . . . . . 9  |-  ( 1Q  e.  Q.  ->  ( /Q `  1Q )  =  1Q )
426, 41ax-mp 5 . . . . . . . 8  |-  ( /Q
`  1Q )  =  1Q
43 mulclpi 9274 . . . . . . . . . . 11  |-  ( ( ( 1st `  x
)  e.  N.  /\  ( 2nd `  x )  e.  N. )  -> 
( ( 1st `  x
)  .N  ( 2nd `  x ) )  e. 
N. )
4432, 34, 43syl2anc 661 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  (
( 1st `  x
)  .N  ( 2nd `  x ) )  e. 
N. )
45 1nqenq 9343 . . . . . . . . . 10  |-  ( ( ( 1st `  x
)  .N  ( 2nd `  x ) )  e. 
N.  ->  1Q  ~Q  <. ( ( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x
)  .N  ( 2nd `  x ) ) >.
)
4644, 45syl 16 . . . . . . . . 9  |-  ( x  e.  Q.  ->  1Q  ~Q 
<. ( ( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x
)  .N  ( 2nd `  x ) ) >.
)
47 elpqn 9306 . . . . . . . . . . 11  |-  ( 1Q  e.  Q.  ->  1Q  e.  ( N.  X.  N. ) )
486, 47ax-mp 5 . . . . . . . . . 10  |-  1Q  e.  ( N.  X.  N. )
49 opelxpi 5021 . . . . . . . . . . 11  |-  ( ( ( ( 1st `  x
)  .N  ( 2nd `  x ) )  e. 
N.  /\  ( ( 1st `  x )  .N  ( 2nd `  x
) )  e.  N. )  ->  <. ( ( 1st `  x )  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x )  .N  ( 2nd `  x ) )
>.  e.  ( N.  X.  N. ) )
5044, 44, 49syl2anc 661 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  <. (
( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x
)  .N  ( 2nd `  x ) ) >.  e.  ( N.  X.  N. ) )
51 nqereq 9316 . . . . . . . . . 10  |-  ( ( 1Q  e.  ( N. 
X.  N. )  /\  <. ( ( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x
)  .N  ( 2nd `  x ) ) >.  e.  ( N.  X.  N. ) )  ->  ( 1Q  ~Q  <. ( ( 1st `  x )  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x )  .N  ( 2nd `  x ) )
>. 
<->  ( /Q `  1Q )  =  ( /Q ` 
<. ( ( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x
)  .N  ( 2nd `  x ) ) >.
) ) )
5248, 50, 51sylancr 663 . . . . . . . . 9  |-  ( x  e.  Q.  ->  ( 1Q  ~Q  <. ( ( 1st `  x )  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x )  .N  ( 2nd `  x ) )
>. 
<->  ( /Q `  1Q )  =  ( /Q ` 
<. ( ( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x
)  .N  ( 2nd `  x ) ) >.
) ) )
5346, 52mpbid 210 . . . . . . . 8  |-  ( x  e.  Q.  ->  ( /Q `  1Q )  =  ( /Q `  <. ( ( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x
)  .N  ( 2nd `  x ) ) >.
) )
5442, 53syl5reqr 2499 . . . . . . 7  |-  ( x  e.  Q.  ->  ( /Q `  <. ( ( 1st `  x )  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x )  .N  ( 2nd `  x ) )
>. )  =  1Q )
5530, 40, 543eqtrd 2488 . . . . . 6  |-  ( x  e.  Q.  ->  (
x  .Q  ( /Q
`  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
)  =  1Q )
56 fvex 5866 . . . . . . 7  |-  ( /Q
`  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )  e.  _V
57 oveq2 6289 . . . . . . . 8  |-  ( y  =  ( /Q `  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )  ->  ( x  .Q  y
)  =  ( x  .Q  ( /Q `  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
) )
5857eqeq1d 2445 . . . . . . 7  |-  ( y  =  ( /Q `  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )  ->  ( ( x  .Q  y )  =  1Q  <->  ( x  .Q  ( /Q
`  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
)  =  1Q ) )
5956, 58spcev 3187 . . . . . 6  |-  ( ( x  .Q  ( /Q
`  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
)  =  1Q  ->  E. y ( x  .Q  y )  =  1Q )
6055, 59syl 16 . . . . 5  |-  ( x  e.  Q.  ->  E. y
( x  .Q  y
)  =  1Q )
61 mulcomnq 9334 . . . . . . 7  |-  ( r  .Q  s )  =  ( s  .Q  r
)
62 mulassnq 9340 . . . . . . 7  |-  ( ( r  .Q  s )  .Q  t )  =  ( r  .Q  (
s  .Q  t ) )
63 mulidnq 9344 . . . . . . 7  |-  ( r  e.  Q.  ->  (
r  .Q  1Q )  =  r )
646, 9, 10, 61, 62, 63caovmo 6497 . . . . . 6  |-  E* y
( x  .Q  y
)  =  1Q
65 eu5 2296 . . . . . 6  |-  ( E! y ( x  .Q  y )  =  1Q  <->  ( E. y ( x  .Q  y )  =  1Q  /\  E* y
( x  .Q  y
)  =  1Q ) )
6664, 65mpbiran2 919 . . . . 5  |-  ( E! y ( x  .Q  y )  =  1Q  <->  E. y ( x  .Q  y )  =  1Q )
6760, 66sylibr 212 . . . 4  |-  ( x  e.  Q.  ->  E! y ( x  .Q  y )  =  1Q )
68 cnvimass 5347 . . . . . . . 8  |-  ( `'  .Q  " { 1Q } )  C_  dom  .Q
69 df-rq 9298 . . . . . . . 8  |-  *Q  =  ( `'  .Q  " { 1Q } )
709eqcomi 2456 . . . . . . . 8  |-  ( Q. 
X.  Q. )  =  dom  .Q
7168, 69, 703sstr4i 3528 . . . . . . 7  |-  *Q  C_  ( Q.  X.  Q. )
72 relxp 5100 . . . . . . 7  |-  Rel  ( Q.  X.  Q. )
73 relss 5080 . . . . . . 7  |-  ( *Q  C_  ( Q.  X.  Q. )  ->  ( Rel  ( Q.  X.  Q. )  ->  Rel  *Q ) )
7471, 72, 73mp2 9 . . . . . 6  |-  Rel  *Q
7569eleq2i 2521 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  *Q  <->  <. x ,  y
>.  e.  ( `'  .Q  " { 1Q } ) )
76 ffn 5721 . . . . . . . . 9  |-  (  .Q  : ( Q.  X.  Q. ) --> Q.  ->  .Q  Fn  ( Q.  X.  Q. )
)
77 fniniseg 5993 . . . . . . . . 9  |-  (  .Q  Fn  ( Q.  X.  Q. )  ->  ( <.
x ,  y >.  e.  ( `'  .Q  " { 1Q } )  <->  ( <. x ,  y >.  e.  ( Q.  X.  Q. )  /\  (  .Q  `  <. x ,  y >. )  =  1Q ) ) )
788, 76, 77mp2b 10 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  ( `'  .Q  " { 1Q } )  <->  ( <. x ,  y >.  e.  ( Q.  X.  Q. )  /\  (  .Q  `  <. x ,  y >. )  =  1Q ) )
79 ancom 450 . . . . . . . . 9  |-  ( (
<. x ,  y >.  e.  ( Q.  X.  Q. )  /\  (  .Q  `  <. x ,  y >.
)  =  1Q )  <-> 
( (  .Q  `  <. x ,  y >.
)  =  1Q  /\  <.
x ,  y >.  e.  ( Q.  X.  Q. ) ) )
80 ancom 450 . . . . . . . . . 10  |-  ( ( x  e.  Q.  /\  ( x  .Q  y
)  =  1Q )  <-> 
( ( x  .Q  y )  =  1Q 
/\  x  e.  Q. ) )
81 eleq1 2515 . . . . . . . . . . . . . . 15  |-  ( ( x  .Q  y )  =  1Q  ->  (
( x  .Q  y
)  e.  Q.  <->  1Q  e.  Q. ) )
826, 81mpbiri 233 . . . . . . . . . . . . . 14  |-  ( ( x  .Q  y )  =  1Q  ->  (
x  .Q  y )  e.  Q. )
839, 10ndmovrcl 6446 . . . . . . . . . . . . . 14  |-  ( ( x  .Q  y )  e.  Q.  ->  (
x  e.  Q.  /\  y  e.  Q. )
)
8482, 83syl 16 . . . . . . . . . . . . 13  |-  ( ( x  .Q  y )  =  1Q  ->  (
x  e.  Q.  /\  y  e.  Q. )
)
85 opelxpi 5021 . . . . . . . . . . . . 13  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  -> 
<. x ,  y >.  e.  ( Q.  X.  Q. ) )
8684, 85syl 16 . . . . . . . . . . . 12  |-  ( ( x  .Q  y )  =  1Q  ->  <. x ,  y >.  e.  ( Q.  X.  Q. )
)
8784simpld 459 . . . . . . . . . . . 12  |-  ( ( x  .Q  y )  =  1Q  ->  x  e.  Q. )
8886, 872thd 240 . . . . . . . . . . 11  |-  ( ( x  .Q  y )  =  1Q  ->  ( <. x ,  y >.  e.  ( Q.  X.  Q. ) 
<->  x  e.  Q. )
)
8988pm5.32i 637 . . . . . . . . . 10  |-  ( ( ( x  .Q  y
)  =  1Q  /\  <.
x ,  y >.  e.  ( Q.  X.  Q. ) )  <->  ( (
x  .Q  y )  =  1Q  /\  x  e.  Q. ) )
90 df-ov 6284 . . . . . . . . . . . 12  |-  ( x  .Q  y )  =  (  .Q  `  <. x ,  y >. )
9190eqeq1i 2450 . . . . . . . . . . 11  |-  ( ( x  .Q  y )  =  1Q  <->  (  .Q  ` 
<. x ,  y >.
)  =  1Q )
9291anbi1i 695 . . . . . . . . . 10  |-  ( ( ( x  .Q  y
)  =  1Q  /\  <.
x ,  y >.  e.  ( Q.  X.  Q. ) )  <->  ( (  .Q  `  <. x ,  y
>. )  =  1Q  /\ 
<. x ,  y >.  e.  ( Q.  X.  Q. ) ) )
9380, 89, 923bitr2ri 274 . . . . . . . . 9  |-  ( ( (  .Q  `  <. x ,  y >. )  =  1Q  /\  <. x ,  y >.  e.  ( Q.  X.  Q. )
)  <->  ( x  e. 
Q.  /\  ( x  .Q  y )  =  1Q ) )
9479, 93bitri 249 . . . . . . . 8  |-  ( (
<. x ,  y >.  e.  ( Q.  X.  Q. )  /\  (  .Q  `  <. x ,  y >.
)  =  1Q )  <-> 
( x  e.  Q.  /\  ( x  .Q  y
)  =  1Q ) )
9575, 78, 943bitri 271 . . . . . . 7  |-  ( <.
x ,  y >.  e.  *Q  <->  ( x  e. 
Q.  /\  ( x  .Q  y )  =  1Q ) )
9695a1i 11 . . . . . 6  |-  ( T. 
->  ( <. x ,  y
>.  e.  *Q  <->  ( x  e.  Q.  /\  ( x  .Q  y )  =  1Q ) ) )
9774, 96opabbi2dv 5142 . . . . 5  |-  ( T. 
->  *Q  =  { <. x ,  y >.  |  ( x  e.  Q.  /\  ( x  .Q  y
)  =  1Q ) } )
9897trud 1392 . . . 4  |-  *Q  =  { <. x ,  y
>.  |  ( x  e.  Q.  /\  ( x  .Q  y )  =  1Q ) }
9918, 20, 67, 98fvopab3g 5937 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  _V )  ->  ( ( *Q `  A )  =  B  <-> 
( A  .Q  B
)  =  1Q ) )
10099ex 434 . 2  |-  ( A  e.  Q.  ->  ( B  e.  _V  ->  ( ( *Q `  A
)  =  B  <->  ( A  .Q  B )  =  1Q ) ) )
1014, 16, 100pm5.21ndd 354 1  |-  ( A  e.  Q.  ->  (
( *Q `  A
)  =  B  <->  ( A  .Q  B )  =  1Q ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383   T. wtru 1384   E.wex 1599    e. wcel 1804   E!weu 2268   E*wmo 2269   _Vcvv 3095    C_ wss 3461   {csn 4014   <.cop 4020   class class class wbr 4437   {copab 4494    X. cxp 4987   `'ccnv 4988   dom cdm 4989   "cima 4992   Rel wrel 4994    Fn wfn 5573   -->wf 5574   ` cfv 5578  (class class class)co 6281   1stc1st 6783   2ndc2nd 6784   N.cnpi 9225    .N cmi 9227    .pQ cmpq 9230    ~Q ceq 9232   Q.cnq 9233   1Qc1q 9234   /Qcerq 9235    .Q cmq 9237   *Qcrq 9238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-omul 7137  df-er 7313  df-ni 9253  df-mi 9255  df-lti 9256  df-mpq 9290  df-enq 9292  df-nq 9293  df-erq 9294  df-mq 9296  df-1nq 9297  df-rq 9298
This theorem is referenced by:  recidnq  9346  recrecnq  9348  reclem3pr  9430
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