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Theorem 1idprl 7147
Description: Lemma for 1idpr 7149. (Contributed by Jim Kingdon, 13-Dec-2019.)
Assertion
Ref Expression
1idprl  |-  ( A  e.  P.  ->  ( 1st `  ( A  .P.  1P ) )  =  ( 1st `  A ) )

Proof of Theorem 1idprl
Dummy variables  x  y  z  w  v  u  f  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 3044 . . . . . 6  |-  ( 1st `  1P )  C_  ( 1st `  1P )
2 rexss 3088 . . . . . 6  |-  ( ( 1st `  1P ) 
C_  ( 1st `  1P )  ->  ( E. g  e.  ( 1st `  1P ) x  =  (
f  .Q  g )  <->  E. g  e.  ( 1st `  1P ) ( g  e.  ( 1st `  1P )  /\  x  =  ( f  .Q  g ) ) ) )
31, 2ax-mp 7 . . . . 5  |-  ( E. g  e.  ( 1st `  1P ) x  =  ( f  .Q  g
)  <->  E. g  e.  ( 1st `  1P ) ( g  e.  ( 1st `  1P )  /\  x  =  ( f  .Q  g ) ) )
4 r19.42v 2524 . . . . . 6  |-  ( E. g  e.  ( 1st `  1P ) ( x 
<Q  f  /\  x  =  ( f  .Q  g ) )  <->  ( x  <Q  f  /\  E. g  e.  ( 1st `  1P ) x  =  (
f  .Q  g ) ) )
5 1pr 7111 . . . . . . . . . . 11  |-  1P  e.  P.
6 prop 7032 . . . . . . . . . . . 12  |-  ( 1P  e.  P.  ->  <. ( 1st `  1P ) ,  ( 2nd `  1P ) >.  e.  P. )
7 elprnql 7038 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  1P ) ,  ( 2nd `  1P ) >.  e.  P.  /\  g  e.  ( 1st `  1P ) )  -> 
g  e.  Q. )
86, 7sylan 277 . . . . . . . . . . 11  |-  ( ( 1P  e.  P.  /\  g  e.  ( 1st `  1P ) )  -> 
g  e.  Q. )
95, 8mpan 415 . . . . . . . . . 10  |-  ( g  e.  ( 1st `  1P )  ->  g  e.  Q. )
10 prop 7032 . . . . . . . . . . . 12  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
11 elprnql 7038 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  f  e.  ( 1st `  A ) )  -> 
f  e.  Q. )
1210, 11sylan 277 . . . . . . . . . . 11  |-  ( ( A  e.  P.  /\  f  e.  ( 1st `  A ) )  -> 
f  e.  Q. )
13 breq1 3848 . . . . . . . . . . . . 13  |-  ( x  =  ( f  .Q  g )  ->  (
x  <Q  f  <->  ( f  .Q  g )  <Q  f
) )
14133ad2ant3 966 . . . . . . . . . . . 12  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  x  =  ( f  .Q  g ) )  -> 
( x  <Q  f  <->  ( f  .Q  g ) 
<Q  f ) )
15 1prl 7112 . . . . . . . . . . . . . . 15  |-  ( 1st `  1P )  =  {
g  |  g  <Q  1Q }
1615abeq2i 2198 . . . . . . . . . . . . . 14  |-  ( g  e.  ( 1st `  1P ) 
<->  g  <Q  1Q )
17 1nq 6923 . . . . . . . . . . . . . . . . 17  |-  1Q  e.  Q.
18 ltmnqg 6958 . . . . . . . . . . . . . . . . 17  |-  ( ( g  e.  Q.  /\  1Q  e.  Q.  /\  f  e.  Q. )  ->  (
g  <Q  1Q  <->  ( f  .Q  g )  <Q  (
f  .Q  1Q ) ) )
1917, 18mp3an2 1261 . . . . . . . . . . . . . . . 16  |-  ( ( g  e.  Q.  /\  f  e.  Q. )  ->  ( g  <Q  1Q  <->  ( f  .Q  g )  <Q  (
f  .Q  1Q ) ) )
2019ancoms 264 . . . . . . . . . . . . . . 15  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( g  <Q  1Q  <->  ( f  .Q  g )  <Q  (
f  .Q  1Q ) ) )
21 mulidnq 6946 . . . . . . . . . . . . . . . . 17  |-  ( f  e.  Q.  ->  (
f  .Q  1Q )  =  f )
2221breq2d 3857 . . . . . . . . . . . . . . . 16  |-  ( f  e.  Q.  ->  (
( f  .Q  g
)  <Q  ( f  .Q  1Q )  <->  ( f  .Q  g )  <Q  f
) )
2322adantr 270 . . . . . . . . . . . . . . 15  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( ( f  .Q  g )  <Q  (
f  .Q  1Q )  <-> 
( f  .Q  g
)  <Q  f ) )
2420, 23bitrd 186 . . . . . . . . . . . . . 14  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( g  <Q  1Q  <->  ( f  .Q  g )  <Q  f
) )
2516, 24syl5rbb 191 . . . . . . . . . . . . 13  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( ( f  .Q  g )  <Q  f  <->  g  e.  ( 1st `  1P ) ) )
26253adant3 963 . . . . . . . . . . . 12  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  x  =  ( f  .Q  g ) )  -> 
( ( f  .Q  g )  <Q  f  <->  g  e.  ( 1st `  1P ) ) )
2714, 26bitrd 186 . . . . . . . . . . 11  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  x  =  ( f  .Q  g ) )  -> 
( x  <Q  f  <->  g  e.  ( 1st `  1P ) ) )
2812, 27syl3an1 1207 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  f  e.  ( 1st `  A ) )  /\  g  e.  Q.  /\  x  =  ( f  .Q  g ) )  -> 
( x  <Q  f  <->  g  e.  ( 1st `  1P ) ) )
299, 28syl3an2 1208 . . . . . . . . 9  |-  ( ( ( A  e.  P.  /\  f  e.  ( 1st `  A ) )  /\  g  e.  ( 1st `  1P )  /\  x  =  ( f  .Q  g ) )  -> 
( x  <Q  f  <->  g  e.  ( 1st `  1P ) ) )
30293expia 1145 . . . . . . . 8  |-  ( ( ( A  e.  P.  /\  f  e.  ( 1st `  A ) )  /\  g  e.  ( 1st `  1P ) )  -> 
( x  =  ( f  .Q  g )  ->  ( x  <Q  f  <-> 
g  e.  ( 1st `  1P ) ) ) )
3130pm5.32rd 439 . . . . . . 7  |-  ( ( ( A  e.  P.  /\  f  e.  ( 1st `  A ) )  /\  g  e.  ( 1st `  1P ) )  -> 
( ( x  <Q  f  /\  x  =  ( f  .Q  g ) )  <->  ( g  e.  ( 1st `  1P )  /\  x  =  ( f  .Q  g ) ) ) )
3231rexbidva 2377 . . . . . 6  |-  ( ( A  e.  P.  /\  f  e.  ( 1st `  A ) )  -> 
( E. g  e.  ( 1st `  1P ) ( x  <Q  f  /\  x  =  ( f  .Q  g ) )  <->  E. g  e.  ( 1st `  1P ) ( g  e.  ( 1st `  1P )  /\  x  =  ( f  .Q  g ) ) ) )
334, 32syl5rbbr 193 . . . . 5  |-  ( ( A  e.  P.  /\  f  e.  ( 1st `  A ) )  -> 
( E. g  e.  ( 1st `  1P ) ( g  e.  ( 1st `  1P )  /\  x  =  ( f  .Q  g ) )  <->  ( x  <Q  f  /\  E. g  e.  ( 1st `  1P ) x  =  (
f  .Q  g ) ) ) )
343, 33syl5bb 190 . . . 4  |-  ( ( A  e.  P.  /\  f  e.  ( 1st `  A ) )  -> 
( E. g  e.  ( 1st `  1P ) x  =  (
f  .Q  g )  <-> 
( x  <Q  f  /\  E. g  e.  ( 1st `  1P ) x  =  ( f  .Q  g ) ) ) )
3534rexbidva 2377 . . 3  |-  ( A  e.  P.  ->  ( E. f  e.  ( 1st `  A ) E. g  e.  ( 1st `  1P ) x  =  ( f  .Q  g
)  <->  E. f  e.  ( 1st `  A ) ( x  <Q  f  /\  E. g  e.  ( 1st `  1P ) x  =  ( f  .Q  g ) ) ) )
36 df-imp 7026 . . . . 5  |-  .P.  =  ( y  e.  P. ,  z  e.  P.  |->  <. { w  e.  Q.  |  E. u  e.  Q.  E. v  e.  Q.  (
u  e.  ( 1st `  y )  /\  v  e.  ( 1st `  z
)  /\  w  =  ( u  .Q  v
) ) } ,  { w  e.  Q.  |  E. u  e.  Q.  E. v  e.  Q.  (
u  e.  ( 2nd `  y )  /\  v  e.  ( 2nd `  z
)  /\  w  =  ( u  .Q  v
) ) } >. )
37 mulclnq 6933 . . . . 5  |-  ( ( u  e.  Q.  /\  v  e.  Q. )  ->  ( u  .Q  v
)  e.  Q. )
3836, 37genpelvl 7069 . . . 4  |-  ( ( A  e.  P.  /\  1P  e.  P. )  -> 
( x  e.  ( 1st `  ( A  .P.  1P ) )  <->  E. f  e.  ( 1st `  A ) E. g  e.  ( 1st `  1P ) x  =  ( f  .Q  g
) ) )
395, 38mpan2 416 . . 3  |-  ( A  e.  P.  ->  (
x  e.  ( 1st `  ( A  .P.  1P ) )  <->  E. f  e.  ( 1st `  A
) E. g  e.  ( 1st `  1P ) x  =  (
f  .Q  g ) ) )
40 prnmaxl 7045 . . . . . . 7  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  x  e.  ( 1st `  A ) )  ->  E. f  e.  ( 1st `  A ) x 
<Q  f )
4110, 40sylan 277 . . . . . 6  |-  ( ( A  e.  P.  /\  x  e.  ( 1st `  A ) )  ->  E. f  e.  ( 1st `  A ) x 
<Q  f )
42 ltrelnq 6922 . . . . . . . . . . . . 13  |-  <Q  C_  ( Q.  X.  Q. )
4342brel 4490 . . . . . . . . . . . 12  |-  ( x 
<Q  f  ->  ( x  e.  Q.  /\  f  e.  Q. ) )
44 ltmnqg 6958 . . . . . . . . . . . . . . . 16  |-  ( ( y  e.  Q.  /\  z  e.  Q.  /\  w  e.  Q. )  ->  (
y  <Q  z  <->  ( w  .Q  y )  <Q  (
w  .Q  z ) ) )
4544adantl 271 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  Q.  /\  f  e.  Q. )  /\  ( y  e.  Q.  /\  z  e.  Q.  /\  w  e.  Q. )
)  ->  ( y  <Q  z  <->  ( w  .Q  y )  <Q  (
w  .Q  z ) ) )
46 simpl 107 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  x  e.  Q. )
47 simpr 108 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  f  e.  Q. )
48 recclnq 6949 . . . . . . . . . . . . . . . 16  |-  ( f  e.  Q.  ->  ( *Q `  f )  e. 
Q. )
4948adantl 271 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( *Q `  f
)  e.  Q. )
50 mulcomnqg 6940 . . . . . . . . . . . . . . . 16  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  z
)  =  ( z  .Q  y ) )
5150adantl 271 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  Q.  /\  f  e.  Q. )  /\  ( y  e.  Q.  /\  z  e.  Q. )
)  ->  ( y  .Q  z )  =  ( z  .Q  y ) )
5245, 46, 47, 49, 51caovord2d 5814 . . . . . . . . . . . . . 14  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( x  <Q  f  <->  ( x  .Q  ( *Q
`  f ) ) 
<Q  ( f  .Q  ( *Q `  f ) ) ) )
53 recidnq 6950 . . . . . . . . . . . . . . . 16  |-  ( f  e.  Q.  ->  (
f  .Q  ( *Q
`  f ) )  =  1Q )
5453breq2d 3857 . . . . . . . . . . . . . . 15  |-  ( f  e.  Q.  ->  (
( x  .Q  ( *Q `  f ) ) 
<Q  ( f  .Q  ( *Q `  f ) )  <-> 
( x  .Q  ( *Q `  f ) ) 
<Q  1Q ) )
5554adantl 271 . . . . . . . . . . . . . 14  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( ( x  .Q  ( *Q `  f ) )  <Q  ( f  .Q  ( *Q `  f
) )  <->  ( x  .Q  ( *Q `  f
) )  <Q  1Q ) )
5652, 55bitrd 186 . . . . . . . . . . . . 13  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( x  <Q  f  <->  ( x  .Q  ( *Q
`  f ) ) 
<Q  1Q ) )
5756biimpd 142 . . . . . . . . . . . 12  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( x  <Q  f  ->  ( x  .Q  ( *Q `  f ) ) 
<Q  1Q ) )
5843, 57mpcom 36 . . . . . . . . . . 11  |-  ( x 
<Q  f  ->  ( x  .Q  ( *Q `  f ) )  <Q  1Q )
59 mulclnq 6933 . . . . . . . . . . . . . 14  |-  ( ( x  e.  Q.  /\  ( *Q `  f )  e.  Q. )  -> 
( x  .Q  ( *Q `  f ) )  e.  Q. )
6048, 59sylan2 280 . . . . . . . . . . . . 13  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( x  .Q  ( *Q `  f ) )  e.  Q. )
6143, 60syl 14 . . . . . . . . . . . 12  |-  ( x 
<Q  f  ->  ( x  .Q  ( *Q `  f ) )  e. 
Q. )
62 breq1 3848 . . . . . . . . . . . . 13  |-  ( g  =  ( x  .Q  ( *Q `  f ) )  ->  ( g  <Q  1Q  <->  ( x  .Q  ( *Q `  f ) )  <Q  1Q )
)
6362, 15elab2g 2762 . . . . . . . . . . . 12  |-  ( ( x  .Q  ( *Q
`  f ) )  e.  Q.  ->  (
( x  .Q  ( *Q `  f ) )  e.  ( 1st `  1P ) 
<->  ( x  .Q  ( *Q `  f ) ) 
<Q  1Q ) )
6461, 63syl 14 . . . . . . . . . . 11  |-  ( x 
<Q  f  ->  ( ( x  .Q  ( *Q
`  f ) )  e.  ( 1st `  1P ) 
<->  ( x  .Q  ( *Q `  f ) ) 
<Q  1Q ) )
6558, 64mpbird 165 . . . . . . . . . 10  |-  ( x 
<Q  f  ->  ( x  .Q  ( *Q `  f ) )  e.  ( 1st `  1P ) )
66 mulassnqg 6941 . . . . . . . . . . . . . 14  |-  ( ( y  e.  Q.  /\  z  e.  Q.  /\  w  e.  Q. )  ->  (
( y  .Q  z
)  .Q  w )  =  ( y  .Q  ( z  .Q  w
) ) )
6766adantl 271 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  Q.  /\  f  e.  Q. )  /\  ( y  e.  Q.  /\  z  e.  Q.  /\  w  e.  Q. )
)  ->  ( (
y  .Q  z )  .Q  w )  =  ( y  .Q  (
z  .Q  w ) ) )
6847, 46, 49, 51, 67caov12d 5826 . . . . . . . . . . . 12  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( f  .Q  (
x  .Q  ( *Q
`  f ) ) )  =  ( x  .Q  ( f  .Q  ( *Q `  f
) ) ) )
6953oveq2d 5668 . . . . . . . . . . . . 13  |-  ( f  e.  Q.  ->  (
x  .Q  ( f  .Q  ( *Q `  f ) ) )  =  ( x  .Q  1Q ) )
7069adantl 271 . . . . . . . . . . . 12  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( x  .Q  (
f  .Q  ( *Q
`  f ) ) )  =  ( x  .Q  1Q ) )
71 mulidnq 6946 . . . . . . . . . . . . 13  |-  ( x  e.  Q.  ->  (
x  .Q  1Q )  =  x )
7271adantr 270 . . . . . . . . . . . 12  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( x  .Q  1Q )  =  x )
7368, 70, 723eqtrrd 2125 . . . . . . . . . . 11  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  x  =  ( f  .Q  ( x  .Q  ( *Q `  f ) ) ) )
7443, 73syl 14 . . . . . . . . . 10  |-  ( x 
<Q  f  ->  x  =  ( f  .Q  (
x  .Q  ( *Q
`  f ) ) ) )
75 oveq2 5660 . . . . . . . . . . . 12  |-  ( g  =  ( x  .Q  ( *Q `  f ) )  ->  ( f  .Q  g )  =  ( f  .Q  ( x  .Q  ( *Q `  f ) ) ) )
7675eqeq2d 2099 . . . . . . . . . . 11  |-  ( g  =  ( x  .Q  ( *Q `  f ) )  ->  ( x  =  ( f  .Q  g )  <->  x  =  ( f  .Q  (
x  .Q  ( *Q
`  f ) ) ) ) )
7776rspcev 2722 . . . . . . . . . 10  |-  ( ( ( x  .Q  ( *Q `  f ) )  e.  ( 1st `  1P )  /\  x  =  ( f  .Q  ( x  .Q  ( *Q `  f ) ) ) )  ->  E. g  e.  ( 1st `  1P ) x  =  (
f  .Q  g ) )
7865, 74, 77syl2anc 403 . . . . . . . . 9  |-  ( x 
<Q  f  ->  E. g  e.  ( 1st `  1P ) x  =  (
f  .Q  g ) )
7978a1i 9 . . . . . . . 8  |-  ( f  e.  ( 1st `  A
)  ->  ( x  <Q  f  ->  E. g  e.  ( 1st `  1P ) x  =  (
f  .Q  g ) ) )
8079ancld 318 . . . . . . 7  |-  ( f  e.  ( 1st `  A
)  ->  ( x  <Q  f  ->  ( x  <Q  f  /\  E. g  e.  ( 1st `  1P ) x  =  (
f  .Q  g ) ) ) )
8180reximia 2468 . . . . . 6  |-  ( E. f  e.  ( 1st `  A ) x  <Q  f  ->  E. f  e.  ( 1st `  A ) ( x  <Q  f  /\  E. g  e.  ( 1st `  1P ) x  =  ( f  .Q  g ) ) )
8241, 81syl 14 . . . . 5  |-  ( ( A  e.  P.  /\  x  e.  ( 1st `  A ) )  ->  E. f  e.  ( 1st `  A ) ( x  <Q  f  /\  E. g  e.  ( 1st `  1P ) x  =  ( f  .Q  g
) ) )
8382ex 113 . . . 4  |-  ( A  e.  P.  ->  (
x  e.  ( 1st `  A )  ->  E. f  e.  ( 1st `  A
) ( x  <Q  f  /\  E. g  e.  ( 1st `  1P ) x  =  (
f  .Q  g ) ) ) )
84 prcdnql 7041 . . . . . . 7  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  f  e.  ( 1st `  A ) )  -> 
( x  <Q  f  ->  x  e.  ( 1st `  A ) ) )
8510, 84sylan 277 . . . . . 6  |-  ( ( A  e.  P.  /\  f  e.  ( 1st `  A ) )  -> 
( x  <Q  f  ->  x  e.  ( 1st `  A ) ) )
8685adantrd 273 . . . . 5  |-  ( ( A  e.  P.  /\  f  e.  ( 1st `  A ) )  -> 
( ( x  <Q  f  /\  E. g  e.  ( 1st `  1P ) x  =  (
f  .Q  g ) )  ->  x  e.  ( 1st `  A ) ) )
8786rexlimdva 2489 . . . 4  |-  ( A  e.  P.  ->  ( E. f  e.  ( 1st `  A ) ( x  <Q  f  /\  E. g  e.  ( 1st `  1P ) x  =  ( f  .Q  g
) )  ->  x  e.  ( 1st `  A
) ) )
8883, 87impbid 127 . . 3  |-  ( A  e.  P.  ->  (
x  e.  ( 1st `  A )  <->  E. f  e.  ( 1st `  A
) ( x  <Q  f  /\  E. g  e.  ( 1st `  1P ) x  =  (
f  .Q  g ) ) ) )
8935, 39, 883bitr4d 218 . 2  |-  ( A  e.  P.  ->  (
x  e.  ( 1st `  ( A  .P.  1P ) )  <->  x  e.  ( 1st `  A ) ) )
9089eqrdv 2086 1  |-  ( A  e.  P.  ->  ( 1st `  ( A  .P.  1P ) )  =  ( 1st `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438   E.wrex 2360    C_ wss 2999   <.cop 3449   class class class wbr 3845   ` cfv 5015  (class class class)co 5652   1stc1st 5909   2ndc2nd 5910   Q.cnq 6837   1Qc1q 6838    .Q cmq 6840   *Qcrq 6841    <Q cltq 6842   P.cnp 6848   1Pc1p 6849    .P. cmp 6851
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-po 4123  df-iso 4124  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-pli 6862  df-mi 6863  df-lti 6864  df-plpq 6901  df-mpq 6902  df-enq 6904  df-nqqs 6905  df-plqqs 6906  df-mqqs 6907  df-1nqqs 6908  df-rq 6909  df-ltnqqs 6910  df-inp 7023  df-i1p 7024  df-imp 7026
This theorem is referenced by:  1idpr  7149
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