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Theorem brecop 6989
Description: Binary relation on a quotient set. Lemma for real number construction. (Contributed by NM, 29-Jan-1996.)
Hypotheses
Ref Expression
brecop.1  |-  .~  e.  _V
brecop.2  |-  .~  Er  ( G  X.  G
)
brecop.4  |-  H  =  ( ( G  X.  G ) /.  .~  )
brecop.5  |-  .<_  =  { <. x ,  y >.  |  ( ( x  e.  H  /\  y  e.  H )  /\  E. z E. w E. v E. u ( ( x  =  [ <. z ,  w >. ]  .~  /\  y  =  [ <. v ,  u >. ]  .~  )  /\  ph ) ) }
brecop.6  |-  ( ( ( ( z  e.  G  /\  w  e.  G )  /\  ( A  e.  G  /\  B  e.  G )
)  /\  ( (
v  e.  G  /\  u  e.  G )  /\  ( C  e.  G  /\  D  e.  G
) ) )  -> 
( ( [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  /\  [ <. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  )  ->  ( ph  <->  ps )
) )
Assertion
Ref Expression
brecop  |-  ( ( ( A  e.  G  /\  B  e.  G
)  /\  ( C  e.  G  /\  D  e.  G ) )  -> 
( [ <. A ,  B >. ]  .~  .<_  [
<. C ,  D >. ]  .~  <->  ps ) )
Distinct variable groups:    x, y,
z, w, v, u, A    x, B, y, z, w, v, u   
x, C, y, z, w, v, u    x, D, y, z, w, v, u    x,  .~ , y, z, w, v, u    x, H, y    z, G, w, v, u    ph, x, y    ps, z, w, v, u
Allowed substitution hints:    ph( z, w, v, u)    ps( x, y)    G( x, y)    H( z, w, v, u)    .<_ ( x, y, z, w, v, u)

Proof of Theorem brecop
StepHypRef Expression
1 brecop.1 . . . 4  |-  .~  e.  _V
2 brecop.4 . . . 4  |-  H  =  ( ( G  X.  G ) /.  .~  )
31, 2ecopqsi 6953 . . 3  |-  ( ( A  e.  G  /\  B  e.  G )  ->  [ <. A ,  B >. ]  .~  e.  H
)
41, 2ecopqsi 6953 . . 3  |-  ( ( C  e.  G  /\  D  e.  G )  ->  [ <. C ,  D >. ]  .~  e.  H
)
5 df-br 4205 . . . . 5  |-  ( [
<. A ,  B >. ]  .~  .<_  [ <. C ,  D >. ]  .~  <->  <. [ <. A ,  B >. ]  .~  ,  [ <. C ,  D >. ]  .~  >.  e.  .<_  )
6 brecop.5 . . . . . 6  |-  .<_  =  { <. x ,  y >.  |  ( ( x  e.  H  /\  y  e.  H )  /\  E. z E. w E. v E. u ( ( x  =  [ <. z ,  w >. ]  .~  /\  y  =  [ <. v ,  u >. ]  .~  )  /\  ph ) ) }
76eleq2i 2499 . . . . 5  |-  ( <. [ <. A ,  B >. ]  .~  ,  [ <. C ,  D >. ]  .~  >.  e.  .<_  <->  <. [
<. A ,  B >. ]  .~  ,  [ <. C ,  D >. ]  .~  >.  e.  { <. x ,  y
>.  |  ( (
x  e.  H  /\  y  e.  H )  /\  E. z E. w E. v E. u ( ( x  =  [ <. z ,  w >. ]  .~  /\  y  =  [ <. v ,  u >. ]  .~  )  /\  ph ) ) } )
85, 7bitri 241 . . . 4  |-  ( [
<. A ,  B >. ]  .~  .<_  [ <. C ,  D >. ]  .~  <->  <. [ <. A ,  B >. ]  .~  ,  [ <. C ,  D >. ]  .~  >.  e.  { <. x ,  y >.  |  ( ( x  e.  H  /\  y  e.  H )  /\  E. z E. w E. v E. u ( ( x  =  [ <. z ,  w >. ]  .~  /\  y  =  [ <. v ,  u >. ]  .~  )  /\  ph ) ) } )
9 eqeq1 2441 . . . . . . . 8  |-  ( x  =  [ <. A ,  B >. ]  .~  ->  ( x  =  [ <. z ,  w >. ]  .~  <->  [
<. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  ) )
109anbi1d 686 . . . . . . 7  |-  ( x  =  [ <. A ,  B >. ]  .~  ->  ( ( x  =  [ <. z ,  w >. ]  .~  /\  y  =  [ <. v ,  u >. ]  .~  )  <->  ( [ <. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\  y  =  [ <. v ,  u >. ]  .~  ) ) )
1110anbi1d 686 . . . . . 6  |-  ( x  =  [ <. A ,  B >. ]  .~  ->  ( ( ( x  =  [ <. z ,  w >. ]  .~  /\  y  =  [ <. v ,  u >. ]  .~  )  /\  ph )  <->  ( ( [
<. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\  y  =  [ <. v ,  u >. ]  .~  )  /\  ph ) ) )
12114exbidv 1640 . . . . 5  |-  ( x  =  [ <. A ,  B >. ]  .~  ->  ( E. z E. w E. v E. u ( ( x  =  [ <. z ,  w >. ]  .~  /\  y  =  [ <. v ,  u >. ]  .~  )  /\  ph )  <->  E. z E. w E. v E. u ( ( [ <. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\  y  =  [ <. v ,  u >. ]  .~  )  /\  ph ) ) )
13 eqeq1 2441 . . . . . . . 8  |-  ( y  =  [ <. C ,  D >. ]  .~  ->  ( y  =  [ <. v ,  u >. ]  .~  <->  [
<. C ,  D >. ]  .~  =  [ <. v ,  u >. ]  .~  ) )
1413anbi2d 685 . . . . . . 7  |-  ( y  =  [ <. C ,  D >. ]  .~  ->  ( ( [ <. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\  y  =  [ <. v ,  u >. ]  .~  )  <->  ( [ <. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\ 
[ <. C ,  D >. ]  .~  =  [ <. v ,  u >. ]  .~  ) ) )
1514anbi1d 686 . . . . . 6  |-  ( y  =  [ <. C ,  D >. ]  .~  ->  ( ( ( [ <. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\  y  =  [ <. v ,  u >. ]  .~  )  /\  ph )  <->  ( ( [
<. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\ 
[ <. C ,  D >. ]  .~  =  [ <. v ,  u >. ]  .~  )  /\  ph ) ) )
16154exbidv 1640 . . . . 5  |-  ( y  =  [ <. C ,  D >. ]  .~  ->  ( E. z E. w E. v E. u ( ( [ <. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\  y  =  [ <. v ,  u >. ]  .~  )  /\  ph )  <->  E. z E. w E. v E. u ( ( [ <. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\  [ <. C ,  D >. ]  .~  =  [ <. v ,  u >. ]  .~  )  /\  ph ) ) )
1712, 16opelopab2 4467 . . . 4  |-  ( ( [ <. A ,  B >. ]  .~  e.  H  /\  [ <. C ,  D >. ]  .~  e.  H
)  ->  ( <. [
<. A ,  B >. ]  .~  ,  [ <. C ,  D >. ]  .~  >.  e.  { <. x ,  y
>.  |  ( (
x  e.  H  /\  y  e.  H )  /\  E. z E. w E. v E. u ( ( x  =  [ <. z ,  w >. ]  .~  /\  y  =  [ <. v ,  u >. ]  .~  )  /\  ph ) ) }  <->  E. z E. w E. v E. u ( ( [
<. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\ 
[ <. C ,  D >. ]  .~  =  [ <. v ,  u >. ]  .~  )  /\  ph ) ) )
188, 17syl5bb 249 . . 3  |-  ( ( [ <. A ,  B >. ]  .~  e.  H  /\  [ <. C ,  D >. ]  .~  e.  H
)  ->  ( [ <. A ,  B >. ]  .~  .<_  [ <. C ,  D >. ]  .~  <->  E. z E. w E. v E. u ( ( [
<. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\ 
[ <. C ,  D >. ]  .~  =  [ <. v ,  u >. ]  .~  )  /\  ph ) ) )
193, 4, 18syl2an 464 . 2  |-  ( ( ( A  e.  G  /\  B  e.  G
)  /\  ( C  e.  G  /\  D  e.  G ) )  -> 
( [ <. A ,  B >. ]  .~  .<_  [
<. C ,  D >. ]  .~  <->  E. z E. w E. v E. u ( ( [ <. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\  [ <. C ,  D >. ]  .~  =  [ <. v ,  u >. ]  .~  )  /\  ph ) ) )
20 opeq12 3978 . . . . . 6  |-  ( ( z  =  A  /\  w  =  B )  -> 
<. z ,  w >.  = 
<. A ,  B >. )
21 eceq1 6933 . . . . . 6  |-  ( <.
z ,  w >.  = 
<. A ,  B >.  ->  [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  )
2220, 21syl 16 . . . . 5  |-  ( ( z  =  A  /\  w  =  B )  ->  [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  )
23 opeq12 3978 . . . . . 6  |-  ( ( v  =  C  /\  u  =  D )  -> 
<. v ,  u >.  = 
<. C ,  D >. )
24 eceq1 6933 . . . . . 6  |-  ( <.
v ,  u >.  = 
<. C ,  D >.  ->  [ <. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  )
2523, 24syl 16 . . . . 5  |-  ( ( v  =  C  /\  u  =  D )  ->  [ <. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  )
2622, 25anim12i 550 . . . 4  |-  ( ( ( z  =  A  /\  w  =  B )  /\  ( v  =  C  /\  u  =  D ) )  -> 
( [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  /\  [ <. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  ) )
27 opelxpi 4902 . . . . . . . 8  |-  ( ( A  e.  G  /\  B  e.  G )  -> 
<. A ,  B >.  e.  ( G  X.  G
) )
28 opelxp 4900 . . . . . . . . 9  |-  ( <.
z ,  w >.  e.  ( G  X.  G
)  <->  ( z  e.  G  /\  w  e.  G ) )
29 brecop.2 . . . . . . . . . . 11  |-  .~  Er  ( G  X.  G
)
3029a1i 11 . . . . . . . . . 10  |-  ( [
<. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  ->  .~  Er  ( G  X.  G ) )
31 id 20 . . . . . . . . . 10  |-  ( [
<. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  ->  [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  )
3230, 31ereldm 6940 . . . . . . . . 9  |-  ( [
<. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  ->  ( <. z ,  w >.  e.  ( G  X.  G )  <->  <. A ,  B >.  e.  ( G  X.  G ) ) )
3328, 32syl5bbr 251 . . . . . . . 8  |-  ( [
<. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  ->  ( ( z  e.  G  /\  w  e.  G )  <->  <. A ,  B >.  e.  ( G  X.  G ) ) )
3427, 33syl5ibr 213 . . . . . . 7  |-  ( [
<. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  ->  ( ( A  e.  G  /\  B  e.  G )  ->  (
z  e.  G  /\  w  e.  G )
) )
35 opelxpi 4902 . . . . . . . 8  |-  ( ( C  e.  G  /\  D  e.  G )  -> 
<. C ,  D >.  e.  ( G  X.  G
) )
36 opelxp 4900 . . . . . . . . 9  |-  ( <.
v ,  u >.  e.  ( G  X.  G
)  <->  ( v  e.  G  /\  u  e.  G ) )
3729a1i 11 . . . . . . . . . 10  |-  ( [
<. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  ->  .~  Er  ( G  X.  G ) )
38 id 20 . . . . . . . . . 10  |-  ( [
<. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  ->  [ <. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  )
3937, 38ereldm 6940 . . . . . . . . 9  |-  ( [
<. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  ->  ( <. v ,  u >.  e.  ( G  X.  G )  <->  <. C ,  D >.  e.  ( G  X.  G ) ) )
4036, 39syl5bbr 251 . . . . . . . 8  |-  ( [
<. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  ->  ( ( v  e.  G  /\  u  e.  G )  <->  <. C ,  D >.  e.  ( G  X.  G ) ) )
4135, 40syl5ibr 213 . . . . . . 7  |-  ( [
<. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  ->  ( ( C  e.  G  /\  D  e.  G )  ->  (
v  e.  G  /\  u  e.  G )
) )
4234, 41im2anan9 809 . . . . . 6  |-  ( ( [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  /\  [ <. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  )  -> 
( ( ( A  e.  G  /\  B  e.  G )  /\  ( C  e.  G  /\  D  e.  G )
)  ->  ( (
z  e.  G  /\  w  e.  G )  /\  ( v  e.  G  /\  u  e.  G
) ) ) )
43 brecop.6 . . . . . . . . 9  |-  ( ( ( ( z  e.  G  /\  w  e.  G )  /\  ( A  e.  G  /\  B  e.  G )
)  /\  ( (
v  e.  G  /\  u  e.  G )  /\  ( C  e.  G  /\  D  e.  G
) ) )  -> 
( ( [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  /\  [ <. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  )  ->  ( ph  <->  ps )
) )
4443an4s 800 . . . . . . . 8  |-  ( ( ( ( z  e.  G  /\  w  e.  G )  /\  (
v  e.  G  /\  u  e.  G )
)  /\  ( ( A  e.  G  /\  B  e.  G )  /\  ( C  e.  G  /\  D  e.  G
) ) )  -> 
( ( [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  /\  [ <. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  )  ->  ( ph  <->  ps )
) )
4544ex 424 . . . . . . 7  |-  ( ( ( z  e.  G  /\  w  e.  G
)  /\  ( v  e.  G  /\  u  e.  G ) )  -> 
( ( ( A  e.  G  /\  B  e.  G )  /\  ( C  e.  G  /\  D  e.  G )
)  ->  ( ( [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  /\  [ <. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  )  -> 
( ph  <->  ps ) ) ) )
4645com13 76 . . . . . 6  |-  ( ( [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  /\  [ <. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  )  -> 
( ( ( A  e.  G  /\  B  e.  G )  /\  ( C  e.  G  /\  D  e.  G )
)  ->  ( (
( z  e.  G  /\  w  e.  G
)  /\  ( v  e.  G  /\  u  e.  G ) )  -> 
( ph  <->  ps ) ) ) )
4742, 46mpdd 38 . . . . 5  |-  ( ( [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  /\  [ <. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  )  -> 
( ( ( A  e.  G  /\  B  e.  G )  /\  ( C  e.  G  /\  D  e.  G )
)  ->  ( ph  <->  ps ) ) )
4847pm5.74d 239 . . . 4  |-  ( ( [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  /\  [ <. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  )  -> 
( ( ( ( A  e.  G  /\  B  e.  G )  /\  ( C  e.  G  /\  D  e.  G
) )  ->  ph )  <->  ( ( ( A  e.  G  /\  B  e.  G )  /\  ( C  e.  G  /\  D  e.  G )
)  ->  ps )
) )
4926, 48cgsex4g 2981 . . 3  |-  ( ( ( A  e.  G  /\  B  e.  G
)  /\  ( C  e.  G  /\  D  e.  G ) )  -> 
( E. z E. w E. v E. u ( ( [
<. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  /\ 
[ <. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  )  /\  (
( ( A  e.  G  /\  B  e.  G )  /\  ( C  e.  G  /\  D  e.  G )
)  ->  ph ) )  <-> 
( ( ( A  e.  G  /\  B  e.  G )  /\  ( C  e.  G  /\  D  e.  G )
)  ->  ps )
) )
50 eqcom 2437 . . . . . . 7  |-  ( [
<. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  <->  [
<. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  )
51 eqcom 2437 . . . . . . 7  |-  ( [
<. C ,  D >. ]  .~  =  [ <. v ,  u >. ]  .~  <->  [
<. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  )
5250, 51anbi12i 679 . . . . . 6  |-  ( ( [ <. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\  [ <. C ,  D >. ]  .~  =  [ <. v ,  u >. ]  .~  )  <->  ( [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  /\ 
[ <. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  ) )
5352a1i 11 . . . . 5  |-  ( ( ( A  e.  G  /\  B  e.  G
)  /\  ( C  e.  G  /\  D  e.  G ) )  -> 
( ( [ <. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\  [ <. C ,  D >. ]  .~  =  [ <. v ,  u >. ]  .~  ) 
<->  ( [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  /\  [ <. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  ) ) )
54 biimt 326 . . . . 5  |-  ( ( ( A  e.  G  /\  B  e.  G
)  /\  ( C  e.  G  /\  D  e.  G ) )  -> 
( ph  <->  ( ( ( A  e.  G  /\  B  e.  G )  /\  ( C  e.  G  /\  D  e.  G
) )  ->  ph )
) )
5553, 54anbi12d 692 . . . 4  |-  ( ( ( A  e.  G  /\  B  e.  G
)  /\  ( C  e.  G  /\  D  e.  G ) )  -> 
( ( ( [
<. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\ 
[ <. C ,  D >. ]  .~  =  [ <. v ,  u >. ]  .~  )  /\  ph ) 
<->  ( ( [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  /\  [ <. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  )  /\  ( ( ( A  e.  G  /\  B  e.  G )  /\  ( C  e.  G  /\  D  e.  G
) )  ->  ph )
) ) )
56554exbidv 1640 . . 3  |-  ( ( ( A  e.  G  /\  B  e.  G
)  /\  ( C  e.  G  /\  D  e.  G ) )  -> 
( E. z E. w E. v E. u ( ( [
<. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\ 
[ <. C ,  D >. ]  .~  =  [ <. v ,  u >. ]  .~  )  /\  ph ) 
<->  E. z E. w E. v E. u ( ( [ <. z ,  w >. ]  .~  =  [ <. A ,  B >. ]  .~  /\  [ <. v ,  u >. ]  .~  =  [ <. C ,  D >. ]  .~  )  /\  ( ( ( A  e.  G  /\  B  e.  G )  /\  ( C  e.  G  /\  D  e.  G
) )  ->  ph )
) ) )
57 biimt 326 . . 3  |-  ( ( ( A  e.  G  /\  B  e.  G
)  /\  ( C  e.  G  /\  D  e.  G ) )  -> 
( ps  <->  ( (
( A  e.  G  /\  B  e.  G
)  /\  ( C  e.  G  /\  D  e.  G ) )  ->  ps ) ) )
5849, 56, 573bitr4d 277 . 2  |-  ( ( ( A  e.  G  /\  B  e.  G
)  /\  ( C  e.  G  /\  D  e.  G ) )  -> 
( E. z E. w E. v E. u ( ( [
<. A ,  B >. ]  .~  =  [ <. z ,  w >. ]  .~  /\ 
[ <. C ,  D >. ]  .~  =  [ <. v ,  u >. ]  .~  )  /\  ph ) 
<->  ps ) )
5919, 58bitrd 245 1  |-  ( ( ( A  e.  G  /\  B  e.  G
)  /\  ( C  e.  G  /\  D  e.  G ) )  -> 
( [ <. A ,  B >. ]  .~  .<_  [
<. C ,  D >. ]  .~  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2948   <.cop 3809   class class class wbr 4204   {copab 4257    X. cxp 4868    Er wer 6894   [cec 6895   /.cqs 6896
This theorem is referenced by:  ltsrpr  8944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4876  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-er 6897  df-ec 6899  df-qs 6903
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