ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  1idprl Unicode version

Theorem 1idprl 6746
Description: Lemma for 1idpr 6748. (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 2992 . . . . . 6  |-  ( 1st `  1P )  C_  ( 1st `  1P )
2 rexss 3035 . . . . . 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 2484 . . . . . 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 6710 . . . . . . . . . . 11  |-  1P  e.  P.
6 prop 6631 . . . . . . . . . . . 12  |-  ( 1P  e.  P.  ->  <. ( 1st `  1P ) ,  ( 2nd `  1P ) >.  e.  P. )
7 elprnql 6637 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  1P ) ,  ( 2nd `  1P ) >.  e.  P.  /\  g  e.  ( 1st `  1P ) )  -> 
g  e.  Q. )
86, 7sylan 271 . . . . . . . . . . 11  |-  ( ( 1P  e.  P.  /\  g  e.  ( 1st `  1P ) )  -> 
g  e.  Q. )
95, 8mpan 408 . . . . . . . . . 10  |-  ( g  e.  ( 1st `  1P )  ->  g  e.  Q. )
10 prop 6631 . . . . . . . . . . . 12  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
11 elprnql 6637 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  f  e.  ( 1st `  A ) )  -> 
f  e.  Q. )
1210, 11sylan 271 . . . . . . . . . . 11  |-  ( ( A  e.  P.  /\  f  e.  ( 1st `  A ) )  -> 
f  e.  Q. )
13 breq1 3795 . . . . . . . . . . . . 13  |-  ( x  =  ( f  .Q  g )  ->  (
x  <Q  f  <->  ( f  .Q  g )  <Q  f
) )
14133ad2ant3 938 . . . . . . . . . . . 12  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  x  =  ( f  .Q  g ) )  -> 
( x  <Q  f  <->  ( f  .Q  g ) 
<Q  f ) )
15 1prl 6711 . . . . . . . . . . . . . . 15  |-  ( 1st `  1P )  =  {
g  |  g  <Q  1Q }
1615abeq2i 2164 . . . . . . . . . . . . . 14  |-  ( g  e.  ( 1st `  1P ) 
<->  g  <Q  1Q )
17 1nq 6522 . . . . . . . . . . . . . . . . 17  |-  1Q  e.  Q.
18 ltmnqg 6557 . . . . . . . . . . . . . . . . 17  |-  ( ( g  e.  Q.  /\  1Q  e.  Q.  /\  f  e.  Q. )  ->  (
g  <Q  1Q  <->  ( f  .Q  g )  <Q  (
f  .Q  1Q ) ) )
1917, 18mp3an2 1231 . . . . . . . . . . . . . . . 16  |-  ( ( g  e.  Q.  /\  f  e.  Q. )  ->  ( g  <Q  1Q  <->  ( f  .Q  g )  <Q  (
f  .Q  1Q ) ) )
2019ancoms 259 . . . . . . . . . . . . . . 15  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( g  <Q  1Q  <->  ( f  .Q  g )  <Q  (
f  .Q  1Q ) ) )
21 mulidnq 6545 . . . . . . . . . . . . . . . . 17  |-  ( f  e.  Q.  ->  (
f  .Q  1Q )  =  f )
2221breq2d 3804 . . . . . . . . . . . . . . . 16  |-  ( f  e.  Q.  ->  (
( f  .Q  g
)  <Q  ( f  .Q  1Q )  <->  ( f  .Q  g )  <Q  f
) )
2322adantr 265 . . . . . . . . . . . . . . 15  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( ( f  .Q  g )  <Q  (
f  .Q  1Q )  <-> 
( f  .Q  g
)  <Q  f ) )
2420, 23bitrd 181 . . . . . . . . . . . . . 14  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( g  <Q  1Q  <->  ( f  .Q  g )  <Q  f
) )
2516, 24syl5rbb 186 . . . . . . . . . . . . 13  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( ( f  .Q  g )  <Q  f  <->  g  e.  ( 1st `  1P ) ) )
26253adant3 935 . . . . . . . . . . . 12  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  x  =  ( f  .Q  g ) )  -> 
( ( f  .Q  g )  <Q  f  <->  g  e.  ( 1st `  1P ) ) )
2714, 26bitrd 181 . . . . . . . . . . 11  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  x  =  ( f  .Q  g ) )  -> 
( x  <Q  f  <->  g  e.  ( 1st `  1P ) ) )
2812, 27syl3an1 1179 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  f  e.  ( 1st `  A ) )  /\  g  e.  Q.  /\  x  =  ( f  .Q  g ) )  -> 
( x  <Q  f  <->  g  e.  ( 1st `  1P ) ) )
299, 28syl3an2 1180 . . . . . . . . 9  |-  ( ( ( A  e.  P.  /\  f  e.  ( 1st `  A ) )  /\  g  e.  ( 1st `  1P )  /\  x  =  ( f  .Q  g ) )  -> 
( x  <Q  f  <->  g  e.  ( 1st `  1P ) ) )
30293expia 1117 . . . . . . . 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 432 . . . . . . 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 2340 . . . . . 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 188 . . . . 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 185 . . . 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 2340 . . 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 6625 . . . . 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 6532 . . . . 5  |-  ( ( u  e.  Q.  /\  v  e.  Q. )  ->  ( u  .Q  v
)  e.  Q. )
3836, 37genpelvl 6668 . . . 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 409 . . 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 6644 . . . . . . 7  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  x  e.  ( 1st `  A ) )  ->  E. f  e.  ( 1st `  A ) x 
<Q  f )
4110, 40sylan 271 . . . . . 6  |-  ( ( A  e.  P.  /\  x  e.  ( 1st `  A ) )  ->  E. f  e.  ( 1st `  A ) x 
<Q  f )
42 ltrelnq 6521 . . . . . . . . . . . . 13  |-  <Q  C_  ( Q.  X.  Q. )
4342brel 4420 . . . . . . . . . . . 12  |-  ( x 
<Q  f  ->  ( x  e.  Q.  /\  f  e.  Q. ) )
44 ltmnqg 6557 . . . . . . . . . . . . . . . 16  |-  ( ( y  e.  Q.  /\  z  e.  Q.  /\  w  e.  Q. )  ->  (
y  <Q  z  <->  ( w  .Q  y )  <Q  (
w  .Q  z ) ) )
4544adantl 266 . . . . . . . . . . . . . . 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 106 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  x  e.  Q. )
47 simpr 107 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  f  e.  Q. )
48 recclnq 6548 . . . . . . . . . . . . . . . 16  |-  ( f  e.  Q.  ->  ( *Q `  f )  e. 
Q. )
4948adantl 266 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( *Q `  f
)  e.  Q. )
50 mulcomnqg 6539 . . . . . . . . . . . . . . . 16  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  z
)  =  ( z  .Q  y ) )
5150adantl 266 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  Q.  /\  f  e.  Q. )  /\  ( y  e.  Q.  /\  z  e.  Q. )
)  ->  ( y  .Q  z )  =  ( z  .Q  y ) )
5245, 46, 47, 49, 51caovord2d 5698 . . . . . . . . . . . . . 14  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( x  <Q  f  <->  ( x  .Q  ( *Q
`  f ) ) 
<Q  ( f  .Q  ( *Q `  f ) ) ) )
53 recidnq 6549 . . . . . . . . . . . . . . . 16  |-  ( f  e.  Q.  ->  (
f  .Q  ( *Q
`  f ) )  =  1Q )
5453breq2d 3804 . . . . . . . . . . . . . . 15  |-  ( f  e.  Q.  ->  (
( x  .Q  ( *Q `  f ) ) 
<Q  ( f  .Q  ( *Q `  f ) )  <-> 
( x  .Q  ( *Q `  f ) ) 
<Q  1Q ) )
5554adantl 266 . . . . . . . . . . . . . 14  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( ( x  .Q  ( *Q `  f ) )  <Q  ( f  .Q  ( *Q `  f
) )  <->  ( x  .Q  ( *Q `  f
) )  <Q  1Q ) )
5652, 55bitrd 181 . . . . . . . . . . . . 13  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( x  <Q  f  <->  ( x  .Q  ( *Q
`  f ) ) 
<Q  1Q ) )
5756biimpd 136 . . . . . . . . . . . 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 6532 . . . . . . . . . . . . . 14  |-  ( ( x  e.  Q.  /\  ( *Q `  f )  e.  Q. )  -> 
( x  .Q  ( *Q `  f ) )  e.  Q. )
6048, 59sylan2 274 . . . . . . . . . . . . 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 3795 . . . . . . . . . . . . 13  |-  ( g  =  ( x  .Q  ( *Q `  f ) )  ->  ( g  <Q  1Q  <->  ( x  .Q  ( *Q `  f ) )  <Q  1Q )
)
6362, 15elab2g 2712 . . . . . . . . . . . 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 160 . . . . . . . . . 10  |-  ( x 
<Q  f  ->  ( x  .Q  ( *Q `  f ) )  e.  ( 1st `  1P ) )
66 mulassnqg 6540 . . . . . . . . . . . . . 14  |-  ( ( y  e.  Q.  /\  z  e.  Q.  /\  w  e.  Q. )  ->  (
( y  .Q  z
)  .Q  w )  =  ( y  .Q  ( z  .Q  w
) ) )
6766adantl 266 . . . . . . . . . . . . 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 5710 . . . . . . . . . . . 12  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( f  .Q  (
x  .Q  ( *Q
`  f ) ) )  =  ( x  .Q  ( f  .Q  ( *Q `  f
) ) ) )
6953oveq2d 5556 . . . . . . . . . . . . 13  |-  ( f  e.  Q.  ->  (
x  .Q  ( f  .Q  ( *Q `  f ) ) )  =  ( x  .Q  1Q ) )
7069adantl 266 . . . . . . . . . . . 12  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( x  .Q  (
f  .Q  ( *Q
`  f ) ) )  =  ( x  .Q  1Q ) )
71 mulidnq 6545 . . . . . . . . . . . . 13  |-  ( x  e.  Q.  ->  (
x  .Q  1Q )  =  x )
7271adantr 265 . . . . . . . . . . . 12  |-  ( ( x  e.  Q.  /\  f  e.  Q. )  ->  ( x  .Q  1Q )  =  x )
7368, 70, 723eqtrrd 2093 . . . . . . . . . . 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 5548 . . . . . . . . . . . 12  |-  ( g  =  ( x  .Q  ( *Q `  f ) )  ->  ( f  .Q  g )  =  ( f  .Q  ( x  .Q  ( *Q `  f ) ) ) )
7675eqeq2d 2067 . . . . . . . . . . 11  |-  ( g  =  ( x  .Q  ( *Q `  f ) )  ->  ( x  =  ( f  .Q  g )  <->  x  =  ( f  .Q  (
x  .Q  ( *Q
`  f ) ) ) ) )
7776rspcev 2673 . . . . . . . . . 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 397 . . . . . . . . 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 312 . . . . . . 7  |-  ( f  e.  ( 1st `  A
)  ->  ( x  <Q  f  ->  ( x  <Q  f  /\  E. g  e.  ( 1st `  1P ) x  =  (
f  .Q  g ) ) ) )
8180reximia 2431 . . . . . 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 112 . . . 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 6640 . . . . . . 7  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  f  e.  ( 1st `  A ) )  -> 
( x  <Q  f  ->  x  e.  ( 1st `  A ) ) )
8510, 84sylan 271 . . . . . 6  |-  ( ( A  e.  P.  /\  f  e.  ( 1st `  A ) )  -> 
( x  <Q  f  ->  x  e.  ( 1st `  A ) ) )
8685adantrd 268 . . . . 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 2450 . . . 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 124 . . 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 213 . 2  |-  ( A  e.  P.  ->  (
x  e.  ( 1st `  ( A  .P.  1P ) )  <->  x  e.  ( 1st `  A ) ) )
9089eqrdv 2054 1  |-  ( A  e.  P.  ->  ( 1st `  ( A  .P.  1P ) )  =  ( 1st `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    /\ w3a 896    = wceq 1259    e. wcel 1409   E.wrex 2324    C_ wss 2945   <.cop 3406   class class class wbr 3792   ` cfv 4930  (class class class)co 5540   1stc1st 5793   2ndc2nd 5794   Q.cnq 6436   1Qc1q 6437    .Q cmq 6439   *Qcrq 6440    <Q cltq 6441   P.cnp 6447   1Pc1p 6448    .P. cmp 6450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-inp 6622  df-i1p 6623  df-imp 6625
This theorem is referenced by:  1idpr  6748
  Copyright terms: Public domain W3C validator