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Theorem recmulnq 8588
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 5539 . . . 4  |-  ( *Q
`  A )  e. 
_V
21a1i 10 . . 3  |-  ( A  e.  Q.  ->  ( *Q `  A )  e. 
_V )
3 eleq1 2343 . . 3  |-  ( ( *Q `  A )  =  B  ->  (
( *Q `  A
)  e.  _V  <->  B  e.  _V ) )
42, 3syl5ibcom 211 . 2  |-  ( A  e.  Q.  ->  (
( *Q `  A
)  =  B  ->  B  e.  _V )
)
5 id 19 . . . . . . 7  |-  ( ( A  .Q  B )  =  1Q  ->  ( A  .Q  B )  =  1Q )
6 1nq 8552 . . . . . . 7  |-  1Q  e.  Q.
75, 6syl6eqel 2371 . . . . . 6  |-  ( ( A  .Q  B )  =  1Q  ->  ( A  .Q  B )  e. 
Q. )
8 mulnqf 8573 . . . . . . . 8  |-  .Q  :
( Q.  X.  Q. )
--> Q.
98fdmi 5394 . . . . . . 7  |-  dom  .Q  =  ( Q.  X.  Q. )
10 0nnq 8548 . . . . . . 7  |-  -.  (/)  e.  Q.
119, 10ndmovrcl 6006 . . . . . 6  |-  ( ( A  .Q  B )  e.  Q.  ->  ( A  e.  Q.  /\  B  e.  Q. ) )
127, 11syl 15 . . . . 5  |-  ( ( A  .Q  B )  =  1Q  ->  ( A  e.  Q.  /\  B  e.  Q. ) )
1312simprd 449 . . . 4  |-  ( ( A  .Q  B )  =  1Q  ->  B  e.  Q. )
14 elex 2796 . . . 4  |-  ( B  e.  Q.  ->  B  e.  _V )
1513, 14syl 15 . . 3  |-  ( ( A  .Q  B )  =  1Q  ->  B  e.  _V )
1615a1i 10 . 2  |-  ( A  e.  Q.  ->  (
( A  .Q  B
)  =  1Q  ->  B  e.  _V ) )
17 oveq1 5865 . . . . 5  |-  ( x  =  A  ->  (
x  .Q  y )  =  ( A  .Q  y ) )
1817eqeq1d 2291 . . . 4  |-  ( x  =  A  ->  (
( x  .Q  y
)  =  1Q  <->  ( A  .Q  y )  =  1Q ) )
19 oveq2 5866 . . . . 5  |-  ( y  =  B  ->  ( A  .Q  y )  =  ( A  .Q  B
) )
2019eqeq1d 2291 . . . 4  |-  ( y  =  B  ->  (
( A  .Q  y
)  =  1Q  <->  ( A  .Q  B )  =  1Q ) )
21 nqerid 8557 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  ( /Q `  x )  =  x )
22 relxp 4794 . . . . . . . . . . . 12  |-  Rel  ( N.  X.  N. )
23 elpqn 8549 . . . . . . . . . . . 12  |-  ( x  e.  Q.  ->  x  e.  ( N.  X.  N. ) )
24 1st2nd 6166 . . . . . . . . . . . 12  |-  ( ( Rel  ( N.  X.  N. )  /\  x  e.  ( N.  X.  N. ) )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
2522, 23, 24sylancr 644 . . . . . . . . . . 11  |-  ( x  e.  Q.  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
2625fveq2d 5529 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  ( /Q `  x )  =  ( /Q `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. ) )
2721, 26eqtr3d 2317 . . . . . . . . 9  |-  ( x  e.  Q.  ->  x  =  ( /Q `  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
)
2827oveq1d 5873 . . . . . . . 8  |-  ( x  e.  Q.  ->  (
x  .Q  ( /Q
`  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
)  =  ( ( /Q `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )  .Q  ( /Q `  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
) )
29 mulerpq 8581 . . . . . . . 8  |-  ( ( /Q `  <. ( 1st `  x ) ,  ( 2nd `  x
) >. )  .Q  ( /Q `  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
)  =  ( /Q
`  ( <. ( 1st `  x ) ,  ( 2nd `  x
) >.  .pQ  <. ( 2nd `  x ) ,  ( 1st `  x )
>. ) )
3028, 29syl6eq 2331 . . . . . . 7  |-  ( x  e.  Q.  ->  (
x  .Q  ( /Q
`  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
)  =  ( /Q
`  ( <. ( 1st `  x ) ,  ( 2nd `  x
) >.  .pQ  <. ( 2nd `  x ) ,  ( 1st `  x )
>. ) ) )
31 xp1st 6149 . . . . . . . . . . 11  |-  ( x  e.  ( N.  X.  N. )  ->  ( 1st `  x )  e.  N. )
3223, 31syl 15 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  ( 1st `  x )  e. 
N. )
33 xp2nd 6150 . . . . . . . . . . 11  |-  ( x  e.  ( N.  X.  N. )  ->  ( 2nd `  x )  e.  N. )
3423, 33syl 15 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  ( 2nd `  x )  e. 
N. )
35 mulpipq 8564 . . . . . . . . . 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 1183 . . . . . . . . 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 8520 . . . . . . . . . 10  |-  ( ( 2nd `  x )  .N  ( 1st `  x
) )  =  ( ( 1st `  x
)  .N  ( 2nd `  x ) )
3837opeq2i 3800 . . . . . . . . 9  |-  <. (
( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 2nd `  x
)  .N  ( 1st `  x ) ) >.  =  <. ( ( 1st `  x )  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x )  .N  ( 2nd `  x ) )
>.
3936, 38syl6eq 2331 . . . . . . . 8  |-  ( x  e.  Q.  ->  ( <. ( 1st `  x
) ,  ( 2nd `  x ) >.  .pQ  <. ( 2nd `  x ) ,  ( 1st `  x
) >. )  =  <. ( ( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x
)  .N  ( 2nd `  x ) ) >.
)
4039fveq2d 5529 . . . . . . 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 8557 . . . . . . . . 9  |-  ( 1Q  e.  Q.  ->  ( /Q `  1Q )  =  1Q )
426, 41ax-mp 8 . . . . . . . 8  |-  ( /Q
`  1Q )  =  1Q
43 mulclpi 8517 . . . . . . . . . . 11  |-  ( ( ( 1st `  x
)  e.  N.  /\  ( 2nd `  x )  e.  N. )  -> 
( ( 1st `  x
)  .N  ( 2nd `  x ) )  e. 
N. )
4432, 34, 43syl2anc 642 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  (
( 1st `  x
)  .N  ( 2nd `  x ) )  e. 
N. )
45 1nqenq 8586 . . . . . . . . . 10  |-  ( ( ( 1st `  x
)  .N  ( 2nd `  x ) )  e. 
N.  ->  1Q  ~Q  <. ( ( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x
)  .N  ( 2nd `  x ) ) >.
)
4644, 45syl 15 . . . . . . . . 9  |-  ( x  e.  Q.  ->  1Q  ~Q 
<. ( ( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x
)  .N  ( 2nd `  x ) ) >.
)
47 elpqn 8549 . . . . . . . . . . 11  |-  ( 1Q  e.  Q.  ->  1Q  e.  ( N.  X.  N. ) )
486, 47ax-mp 8 . . . . . . . . . 10  |-  1Q  e.  ( N.  X.  N. )
49 opelxpi 4721 . . . . . . . . . . 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 642 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  <. (
( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x
)  .N  ( 2nd `  x ) ) >.  e.  ( N.  X.  N. ) )
51 nqereq 8559 . . . . . . . . . 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 644 . . . . . . . . 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 201 . . . . . . . 8  |-  ( x  e.  Q.  ->  ( /Q `  1Q )  =  ( /Q `  <. ( ( 1st `  x
)  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x
)  .N  ( 2nd `  x ) ) >.
) )
5442, 53syl5reqr 2330 . . . . . . 7  |-  ( x  e.  Q.  ->  ( /Q `  <. ( ( 1st `  x )  .N  ( 2nd `  x ) ) ,  ( ( 1st `  x )  .N  ( 2nd `  x ) )
>. )  =  1Q )
5530, 40, 543eqtrd 2319 . . . . . 6  |-  ( x  e.  Q.  ->  (
x  .Q  ( /Q
`  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
)  =  1Q )
56 fvex 5539 . . . . . . 7  |-  ( /Q
`  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )  e.  _V
57 oveq2 5866 . . . . . . . 8  |-  ( y  =  ( /Q `  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )  ->  ( x  .Q  y
)  =  ( x  .Q  ( /Q `  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
) )
5857eqeq1d 2291 . . . . . . 7  |-  ( y  =  ( /Q `  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )  ->  ( ( x  .Q  y )  =  1Q  <->  ( x  .Q  ( /Q
`  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
)  =  1Q ) )
5956, 58spcev 2875 . . . . . 6  |-  ( ( x  .Q  ( /Q
`  <. ( 2nd `  x
) ,  ( 1st `  x ) >. )
)  =  1Q  ->  E. y ( x  .Q  y )  =  1Q )
6055, 59syl 15 . . . . 5  |-  ( x  e.  Q.  ->  E. y
( x  .Q  y
)  =  1Q )
61 mulcomnq 8577 . . . . . . 7  |-  ( r  .Q  s )  =  ( s  .Q  r
)
62 mulassnq 8583 . . . . . . 7  |-  ( ( r  .Q  s )  .Q  t )  =  ( r  .Q  (
s  .Q  t ) )
63 mulidnq 8587 . . . . . . 7  |-  ( r  e.  Q.  ->  (
r  .Q  1Q )  =  r )
646, 9, 10, 61, 62, 63caovmo 6057 . . . . . 6  |-  E* y
( x  .Q  y
)  =  1Q
65 eu5 2181 . . . . . 6  |-  ( E! y ( x  .Q  y )  =  1Q  <->  ( E. y ( x  .Q  y )  =  1Q  /\  E* y
( x  .Q  y
)  =  1Q ) )
6664, 65mpbiran2 885 . . . . 5  |-  ( E! y ( x  .Q  y )  =  1Q  <->  E. y ( x  .Q  y )  =  1Q )
6760, 66sylibr 203 . . . 4  |-  ( x  e.  Q.  ->  E! y ( x  .Q  y )  =  1Q )
68 cnvimass 5033 . . . . . . . 8  |-  ( `'  .Q  " { 1Q } )  C_  dom  .Q
69 df-rq 8541 . . . . . . . 8  |-  *Q  =  ( `'  .Q  " { 1Q } )
709eqcomi 2287 . . . . . . . 8  |-  ( Q. 
X.  Q. )  =  dom  .Q
7168, 69, 703sstr4i 3217 . . . . . . 7  |-  *Q  C_  ( Q.  X.  Q. )
72 relxp 4794 . . . . . . 7  |-  Rel  ( Q.  X.  Q. )
73 relss 4775 . . . . . . 7  |-  ( *Q  C_  ( Q.  X.  Q. )  ->  ( Rel  ( Q.  X.  Q. )  ->  Rel  *Q ) )
7471, 72, 73mp2 17 . . . . . 6  |-  Rel  *Q
7569eleq2i 2347 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  *Q  <->  <. x ,  y
>.  e.  ( `'  .Q  " { 1Q } ) )
76 ffn 5389 . . . . . . . . 9  |-  (  .Q  : ( Q.  X.  Q. ) --> Q.  ->  .Q  Fn  ( Q.  X.  Q. )
)
77 fniniseg 5646 . . . . . . . . 9  |-  (  .Q  Fn  ( Q.  X.  Q. )  ->  ( <.
x ,  y >.  e.  ( `'  .Q  " { 1Q } )  <->  ( <. x ,  y >.  e.  ( Q.  X.  Q. )  /\  (  .Q  `  <. x ,  y >. )  =  1Q ) ) )
788, 76, 77mp2b 9 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  ( `'  .Q  " { 1Q } )  <->  ( <. x ,  y >.  e.  ( Q.  X.  Q. )  /\  (  .Q  `  <. x ,  y >. )  =  1Q ) )
79 ancom 437 . . . . . . . . 9  |-  ( (
<. x ,  y >.  e.  ( Q.  X.  Q. )  /\  (  .Q  `  <. x ,  y >.
)  =  1Q )  <-> 
( (  .Q  `  <. x ,  y >.
)  =  1Q  /\  <.
x ,  y >.  e.  ( Q.  X.  Q. ) ) )
80 ancom 437 . . . . . . . . . 10  |-  ( ( x  e.  Q.  /\  ( x  .Q  y
)  =  1Q )  <-> 
( ( x  .Q  y )  =  1Q 
/\  x  e.  Q. ) )
81 eleq1 2343 . . . . . . . . . . . . . . 15  |-  ( ( x  .Q  y )  =  1Q  ->  (
( x  .Q  y
)  e.  Q.  <->  1Q  e.  Q. ) )
826, 81mpbiri 224 . . . . . . . . . . . . . 14  |-  ( ( x  .Q  y )  =  1Q  ->  (
x  .Q  y )  e.  Q. )
839, 10ndmovrcl 6006 . . . . . . . . . . . . . 14  |-  ( ( x  .Q  y )  e.  Q.  ->  (
x  e.  Q.  /\  y  e.  Q. )
)
8482, 83syl 15 . . . . . . . . . . . . 13  |-  ( ( x  .Q  y )  =  1Q  ->  (
x  e.  Q.  /\  y  e.  Q. )
)
85 opelxpi 4721 . . . . . . . . . . . . 13  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  -> 
<. x ,  y >.  e.  ( Q.  X.  Q. ) )
8684, 85syl 15 . . . . . . . . . . . 12  |-  ( ( x  .Q  y )  =  1Q  ->  <. x ,  y >.  e.  ( Q.  X.  Q. )
)
8784simpld 445 . . . . . . . . . . . 12  |-  ( ( x  .Q  y )  =  1Q  ->  x  e.  Q. )
8886, 872thd 231 . . . . . . . . . . 11  |-  ( ( x  .Q  y )  =  1Q  ->  ( <. x ,  y >.  e.  ( Q.  X.  Q. ) 
<->  x  e.  Q. )
)
8988pm5.32i 618 . . . . . . . . . 10  |-  ( ( ( x  .Q  y
)  =  1Q  /\  <.
x ,  y >.  e.  ( Q.  X.  Q. ) )  <->  ( (
x  .Q  y )  =  1Q  /\  x  e.  Q. ) )
90 df-ov 5861 . . . . . . . . . . . 12  |-  ( x  .Q  y )  =  (  .Q  `  <. x ,  y >. )
9190eqeq1i 2290 . . . . . . . . . . 11  |-  ( ( x  .Q  y )  =  1Q  <->  (  .Q  ` 
<. x ,  y >.
)  =  1Q )
9291anbi1i 676 . . . . . . . . . 10  |-  ( ( ( x  .Q  y
)  =  1Q  /\  <.
x ,  y >.  e.  ( Q.  X.  Q. ) )  <->  ( (  .Q  `  <. x ,  y
>. )  =  1Q  /\ 
<. x ,  y >.  e.  ( Q.  X.  Q. ) ) )
9380, 89, 923bitr2ri 265 . . . . . . . . 9  |-  ( ( (  .Q  `  <. x ,  y >. )  =  1Q  /\  <. x ,  y >.  e.  ( Q.  X.  Q. )
)  <->  ( x  e. 
Q.  /\  ( x  .Q  y )  =  1Q ) )
9479, 93bitri 240 . . . . . . . 8  |-  ( (
<. x ,  y >.  e.  ( Q.  X.  Q. )  /\  (  .Q  `  <. x ,  y >.
)  =  1Q )  <-> 
( x  e.  Q.  /\  ( x  .Q  y
)  =  1Q ) )
9575, 78, 943bitri 262 . . . . . . 7  |-  ( <.
x ,  y >.  e.  *Q  <->  ( x  e. 
Q.  /\  ( x  .Q  y )  =  1Q ) )
9695a1i 10 . . . . . 6  |-  (  T. 
->  ( <. x ,  y
>.  e.  *Q  <->  ( x  e.  Q.  /\  ( x  .Q  y )  =  1Q ) ) )
9774, 96opabbi2dv 4833 . . . . 5  |-  (  T. 
->  *Q  =  { <. x ,  y >.  |  ( x  e.  Q.  /\  ( x  .Q  y
)  =  1Q ) } )
9897trud 1314 . . . 4  |-  *Q  =  { <. x ,  y
>.  |  ( x  e.  Q.  /\  ( x  .Q  y )  =  1Q ) }
9918, 20, 67, 98fvopab3g 5598 . . 3  |-  ( ( A  e.  Q.  /\  B  e.  _V )  ->  ( ( *Q `  A )  =  B  <-> 
( A  .Q  B
)  =  1Q ) )
10099ex 423 . 2  |-  ( A  e.  Q.  ->  ( B  e.  _V  ->  ( ( *Q `  A
)  =  B  <->  ( A  .Q  B )  =  1Q ) ) )
1014, 16, 100pm5.21ndd 343 1  |-  ( A  e.  Q.  ->  (
( *Q `  A
)  =  B  <->  ( A  .Q  B )  =  1Q ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    T. wtru 1307   E.wex 1528    = wceq 1623    e. wcel 1684   E!weu 2143   E*wmo 2144   _Vcvv 2788    C_ wss 3152   {csn 3640   <.cop 3643   class class class wbr 4023   {copab 4076    X. cxp 4687   `'ccnv 4688   dom cdm 4689   "cima 4692   Rel wrel 4694    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   N.cnpi 8466    .N cmi 8468    .pQ cmpq 8471    ~Q ceq 8473   Q.cnq 8474   1Qc1q 8475   /Qcerq 8476    .Q cmq 8478   *Qcrq 8479
This theorem is referenced by:  recidnq  8589  recrecnq  8591  reclem3pr  8673
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-omul 6484  df-er 6660  df-ni 8496  df-mi 8498  df-lti 8499  df-mpq 8533  df-enq 8535  df-nq 8536  df-erq 8537  df-mq 8539  df-1nq 8540  df-rq 8541
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