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Theorem 1idprl 7210
Description: Lemma for 1idpr 7212. (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 3045 . . . . . 6  |-  ( 1st `  1P )  C_  ( 1st `  1P )
2 rexss 3089 . . . . . 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 2525 . . . . . 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 7174 . . . . . . . . . . 11  |-  1P  e.  P.
6 prop 7095 . . . . . . . . . . . 12  |-  ( 1P  e.  P.  ->  <. ( 1st `  1P ) ,  ( 2nd `  1P ) >.  e.  P. )
7 elprnql 7101 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  1P ) ,  ( 2nd `  1P ) >.  e.  P.  /\  g  e.  ( 1st `  1P ) )  -> 
g  e.  Q. )
86, 7sylan 278 . . . . . . . . . . 11  |-  ( ( 1P  e.  P.  /\  g  e.  ( 1st `  1P ) )  -> 
g  e.  Q. )
95, 8mpan 416 . . . . . . . . . 10  |-  ( g  e.  ( 1st `  1P )  ->  g  e.  Q. )
10 prop 7095 . . . . . . . . . . . 12  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
11 elprnql 7101 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  f  e.  ( 1st `  A ) )  -> 
f  e.  Q. )
1210, 11sylan 278 . . . . . . . . . . 11  |-  ( ( A  e.  P.  /\  f  e.  ( 1st `  A ) )  -> 
f  e.  Q. )
13 breq1 3854 . . . . . . . . . . . . 13  |-  ( x  =  ( f  .Q  g )  ->  (
x  <Q  f  <->  ( f  .Q  g )  <Q  f
) )
14133ad2ant3 967 . . . . . . . . . . . 12  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  x  =  ( f  .Q  g ) )  -> 
( x  <Q  f  <->  ( f  .Q  g ) 
<Q  f ) )
15 1prl 7175 . . . . . . . . . . . . . . 15  |-  ( 1st `  1P )  =  {
g  |  g  <Q  1Q }
1615abeq2i 2199 . . . . . . . . . . . . . 14  |-  ( g  e.  ( 1st `  1P ) 
<->  g  <Q  1Q )
17 1nq 6986 . . . . . . . . . . . . . . . . 17  |-  1Q  e.  Q.
18 ltmnqg 7021 . . . . . . . . . . . . . . . . 17  |-  ( ( g  e.  Q.  /\  1Q  e.  Q.  /\  f  e.  Q. )  ->  (
g  <Q  1Q  <->  ( f  .Q  g )  <Q  (
f  .Q  1Q ) ) )
1917, 18mp3an2 1262 . . . . . . . . . . . . . . . 16  |-  ( ( g  e.  Q.  /\  f  e.  Q. )  ->  ( g  <Q  1Q  <->  ( f  .Q  g )  <Q  (
f  .Q  1Q ) ) )
2019ancoms 265 . . . . . . . . . . . . . . 15  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( g  <Q  1Q  <->  ( f  .Q  g )  <Q  (
f  .Q  1Q ) ) )
21 mulidnq 7009 . . . . . . . . . . . . . . . . 17  |-  ( f  e.  Q.  ->  (
f  .Q  1Q )  =  f )
2221breq2d 3863 . . . . . . . . . . . . . . . 16  |-  ( f  e.  Q.  ->  (
( f  .Q  g
)  <Q  ( f  .Q  1Q )  <->  ( f  .Q  g )  <Q  f
) )
2322adantr 271 . . . . . . . . . . . . . . 15  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( ( f  .Q  g )  <Q  (
f  .Q  1Q )  <-> 
( f  .Q  g
)  <Q  f ) )
2420, 23bitrd 187 . . . . . . . . . . . . . 14  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( g  <Q  1Q  <->  ( f  .Q  g )  <Q  f
) )
2516, 24syl5rbb 192 . . . . . . . . . . . . 13  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( ( f  .Q  g )  <Q  f  <->  g  e.  ( 1st `  1P ) ) )
26253adant3 964 . . . . . . . . . . . 12  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  x  =  ( f  .Q  g ) )  -> 
( ( f  .Q  g )  <Q  f  <->  g  e.  ( 1st `  1P ) ) )
2714, 26bitrd 187 . . . . . . . . . . 11  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  x  =  ( f  .Q  g ) )  -> 
( x  <Q  f  <->  g  e.  ( 1st `  1P ) ) )
2812, 27syl3an1 1208 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  f  e.  ( 1st `  A ) )  /\  g  e.  Q.  /\  x  =  ( f  .Q  g ) )  -> 
( x  <Q  f  <->  g  e.  ( 1st `  1P ) ) )
299, 28syl3an2 1209 . . . . . . . . 9  |-  ( ( ( A  e.  P.  /\  f  e.  ( 1st `  A ) )  /\  g  e.  ( 1st `  1P )  /\  x  =  ( f  .Q  g ) )  -> 
( x  <Q  f  <->  g  e.  ( 1st `  1P ) ) )
30293expia 1146 . . . . . . . 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 440 . . . . . . 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 2378 . . . . . 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 194 . . . . 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 191 . . . 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 2378 . . 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 7089 . . . . 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 6996 . . . . 5  |-  ( ( u  e.  Q.  /\  v  e.  Q. )  ->  ( u  .Q  v
)  e.  Q. )
3836, 37genpelvl 7132 . . . 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 417 . . 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 7108 . . . . . . 7  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  x  e.  ( 1st `  A ) )  ->  E. f  e.  ( 1st `  A ) x 
<Q  f )
4110, 40sylan 278 . . . . . 6  |-  ( ( A  e.  P.  /\  x  e.  ( 1st `  A ) )  ->  E. f  e.  ( 1st `  A ) x 
<Q  f )
42 ltrelnq 6985 . . . . . . . . . . . . 13  |-  <Q  C_  ( Q.  X.  Q. )
4342brel 4503 . . . . . . . . . . . 12  |-  ( x 
<Q  f  ->  ( x  e.  Q.  /\  f  e.  Q. ) )
44 ltmnqg 7021 . . . . . . . . . . . . . . . 16  |-  ( ( y  e.  Q.  /\  z  e.  Q.  /\  w  e.  Q. )  ->  (
y  <Q  z  <->  ( w  .Q  y )  <Q  (
w  .Q  z ) ) )
4544adantl 272 . . . . . . . . . . . . . . 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 108 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  x  e.  Q. )
47 simpr 109 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  f  e.  Q. )
48 recclnq 7012 . . . . . . . . . . . . . . . 16  |-  ( f  e.  Q.  ->  ( *Q `  f )  e. 
Q. )
4948adantl 272 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( *Q `  f
)  e.  Q. )
50 mulcomnqg 7003 . . . . . . . . . . . . . . . 16  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  z
)  =  ( z  .Q  y ) )
5150adantl 272 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  Q.  /\  f  e.  Q. )  /\  ( y  e.  Q.  /\  z  e.  Q. )
)  ->  ( y  .Q  z )  =  ( z  .Q  y ) )
5245, 46, 47, 49, 51caovord2d 5828 . . . . . . . . . . . . . 14  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( x  <Q  f  <->  ( x  .Q  ( *Q
`  f ) ) 
<Q  ( f  .Q  ( *Q `  f ) ) ) )
53 recidnq 7013 . . . . . . . . . . . . . . . 16  |-  ( f  e.  Q.  ->  (
f  .Q  ( *Q
`  f ) )  =  1Q )
5453breq2d 3863 . . . . . . . . . . . . . . 15  |-  ( f  e.  Q.  ->  (
( x  .Q  ( *Q `  f ) ) 
<Q  ( f  .Q  ( *Q `  f ) )  <-> 
( x  .Q  ( *Q `  f ) ) 
<Q  1Q ) )
5554adantl 272 . . . . . . . . . . . . . 14  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( ( x  .Q  ( *Q `  f ) )  <Q  ( f  .Q  ( *Q `  f
) )  <->  ( x  .Q  ( *Q `  f
) )  <Q  1Q ) )
5652, 55bitrd 187 . . . . . . . . . . . . 13  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( x  <Q  f  <->  ( x  .Q  ( *Q
`  f ) ) 
<Q  1Q ) )
5756biimpd 143 . . . . . . . . . . . 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 6996 . . . . . . . . . . . . . 14  |-  ( ( x  e.  Q.  /\  ( *Q `  f )  e.  Q. )  -> 
( x  .Q  ( *Q `  f ) )  e.  Q. )
6048, 59sylan2 281 . . . . . . . . . . . . 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 3854 . . . . . . . . . . . . 13  |-  ( g  =  ( x  .Q  ( *Q `  f ) )  ->  ( g  <Q  1Q  <->  ( x  .Q  ( *Q `  f ) )  <Q  1Q )
)
6362, 15elab2g 2763 . . . . . . . . . . . 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 166 . . . . . . . . . 10  |-  ( x 
<Q  f  ->  ( x  .Q  ( *Q `  f ) )  e.  ( 1st `  1P ) )
66 mulassnqg 7004 . . . . . . . . . . . . . 14  |-  ( ( y  e.  Q.  /\  z  e.  Q.  /\  w  e.  Q. )  ->  (
( y  .Q  z
)  .Q  w )  =  ( y  .Q  ( z  .Q  w
) ) )
6766adantl 272 . . . . . . . . . . . . 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 5840 . . . . . . . . . . . 12  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( f  .Q  (
x  .Q  ( *Q
`  f ) ) )  =  ( x  .Q  ( f  .Q  ( *Q `  f
) ) ) )
6953oveq2d 5682 . . . . . . . . . . . . 13  |-  ( f  e.  Q.  ->  (
x  .Q  ( f  .Q  ( *Q `  f ) ) )  =  ( x  .Q  1Q ) )
7069adantl 272 . . . . . . . . . . . 12  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( x  .Q  (
f  .Q  ( *Q
`  f ) ) )  =  ( x  .Q  1Q ) )
71 mulidnq 7009 . . . . . . . . . . . . 13  |-  ( x  e.  Q.  ->  (
x  .Q  1Q )  =  x )
7271adantr 271 . . . . . . . . . . . 12  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( x  .Q  1Q )  =  x )
7368, 70, 723eqtrrd 2126 . . . . . . . . . . 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 5674 . . . . . . . . . . . 12  |-  ( g  =  ( x  .Q  ( *Q `  f ) )  ->  ( f  .Q  g )  =  ( f  .Q  ( x  .Q  ( *Q `  f ) ) ) )
7675eqeq2d 2100 . . . . . . . . . . 11  |-  ( g  =  ( x  .Q  ( *Q `  f ) )  ->  ( x  =  ( f  .Q  g )  <->  x  =  ( f  .Q  (
x  .Q  ( *Q
`  f ) ) ) ) )
7776rspcev 2723 . . . . . . . . . 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 404 . . . . . . . . 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 319 . . . . . . 7  |-  ( f  e.  ( 1st `  A
)  ->  ( x  <Q  f  ->  ( x  <Q  f  /\  E. g  e.  ( 1st `  1P ) x  =  (
f  .Q  g ) ) ) )
8180reximia 2469 . . . . . 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 114 . . . 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 7104 . . . . . . 7  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  f  e.  ( 1st `  A ) )  -> 
( x  <Q  f  ->  x  e.  ( 1st `  A ) ) )
8510, 84sylan 278 . . . . . 6  |-  ( ( A  e.  P.  /\  f  e.  ( 1st `  A ) )  -> 
( x  <Q  f  ->  x  e.  ( 1st `  A ) ) )
8685adantrd 274 . . . . 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 2490 . . . 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 128 . . 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 219 . 2  |-  ( A  e.  P.  ->  (
x  e.  ( 1st `  ( A  .P.  1P ) )  <->  x  e.  ( 1st `  A ) ) )
9089eqrdv 2087 1  |-  ( A  e.  P.  ->  ( 1st `  ( A  .P.  1P ) )  =  ( 1st `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 925    = wceq 1290    e. wcel 1439   E.wrex 2361    C_ wss 3000   <.cop 3453   class class class wbr 3851   ` cfv 5028  (class class class)co 5666   1stc1st 5923   2ndc2nd 5924   Q.cnq 6900   1Qc1q 6901    .Q cmq 6903   *Qcrq 6904    <Q cltq 6905   P.cnp 6911   1Pc1p 6912    .P. cmp 6914
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-eprel 4125  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-1o 6195  df-oadd 6199  df-omul 6200  df-er 6306  df-ec 6308  df-qs 6312  df-ni 6924  df-pli 6925  df-mi 6926  df-lti 6927  df-plpq 6964  df-mpq 6965  df-enq 6967  df-nqqs 6968  df-plqqs 6969  df-mqqs 6970  df-1nqqs 6971  df-rq 6972  df-ltnqqs 6973  df-inp 7086  df-i1p 7087  df-imp 7089
This theorem is referenced by:  1idpr  7212
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