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Theorem xpcco 15299
Description: Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
xpccofval.t  |-  T  =  ( C  X.c  D )
xpccofval.b  |-  B  =  ( Base `  T
)
xpccofval.k  |-  K  =  ( Hom  `  T
)
xpccofval.o1  |-  .x.  =  (comp `  C )
xpccofval.o2  |-  .xb  =  (comp `  D )
xpccofval.o  |-  O  =  (comp `  T )
xpcco.x  |-  ( ph  ->  X  e.  B )
xpcco.y  |-  ( ph  ->  Y  e.  B )
xpcco.z  |-  ( ph  ->  Z  e.  B )
xpcco.f  |-  ( ph  ->  F  e.  ( X K Y ) )
xpcco.g  |-  ( ph  ->  G  e.  ( Y K Z ) )
Assertion
Ref Expression
xpcco  |-  ( ph  ->  ( G ( <. X ,  Y >. O Z ) F )  =  <. ( ( 1st `  G ) ( <.
( 1st `  X
) ,  ( 1st `  Y ) >.  .x.  ( 1st `  Z ) ) ( 1st `  F
) ) ,  ( ( 2nd `  G
) ( <. ( 2nd `  X ) ,  ( 2nd `  Y
) >.  .xb  ( 2nd `  Z
) ) ( 2nd `  F ) ) >.
)

Proof of Theorem xpcco
Dummy variables  f 
g  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpccofval.t . . 3  |-  T  =  ( C  X.c  D )
2 xpccofval.b . . 3  |-  B  =  ( Base `  T
)
3 xpccofval.k . . 3  |-  K  =  ( Hom  `  T
)
4 xpccofval.o1 . . 3  |-  .x.  =  (comp `  C )
5 xpccofval.o2 . . 3  |-  .xb  =  (comp `  D )
6 xpccofval.o . . 3  |-  O  =  (comp `  T )
71, 2, 3, 4, 5, 6xpccofval 15298 . 2  |-  O  =  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x
) K y ) ,  f  e.  ( K `  x ) 
|->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>.  .x.  ( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) ) >.
) )
8 xpcco.x . . . 4  |-  ( ph  ->  X  e.  B )
9 xpcco.y . . . 4  |-  ( ph  ->  Y  e.  B )
10 opelxpi 5023 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
118, 9, 10syl2anc 661 . . 3  |-  ( ph  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
12 xpcco.z . . . 4  |-  ( ph  ->  Z  e.  B )
1312adantr 465 . . 3  |-  ( (
ph  /\  x  =  <. X ,  Y >. )  ->  Z  e.  B
)
14 ovex 6300 . . . . 5  |-  ( ( 2nd `  x ) K y )  e. 
_V
15 fvex 5867 . . . . 5  |-  ( K `
 x )  e. 
_V
1614, 15mpt2ex 6850 . . . 4  |-  ( g  e.  ( ( 2nd `  x ) K y ) ,  f  e.  ( K `  x
)  |->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>.  .x.  ( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) ) >.
)  e.  _V
1716a1i 11 . . 3  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  (
g  e.  ( ( 2nd `  x ) K y ) ,  f  e.  ( K `
 x )  |->  <.
( ( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.  .x.  ( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) ) >.
)  e.  _V )
18 xpcco.g . . . . . 6  |-  ( ph  ->  G  e.  ( Y K Z ) )
1918adantr 465 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  G  e.  ( Y K Z ) )
20 simprl 755 . . . . . . . 8  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  x  =  <. X ,  Y >. )
2120fveq2d 5861 . . . . . . 7  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  ( 2nd `  x )  =  ( 2nd `  <. X ,  Y >. )
)
22 op2ndg 6787 . . . . . . . . 9  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
238, 9, 22syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
2423adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
2521, 24eqtrd 2501 . . . . . 6  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  ( 2nd `  x )  =  Y )
26 simprr 756 . . . . . 6  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  y  =  Z )
2725, 26oveq12d 6293 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  (
( 2nd `  x
) K y )  =  ( Y K Z ) )
2819, 27eleqtrrd 2551 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  G  e.  ( ( 2nd `  x
) K y ) )
29 xpcco.f . . . . . . 7  |-  ( ph  ->  F  e.  ( X K Y ) )
3029adantr 465 . . . . . 6  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  F  e.  ( X K Y ) )
3120fveq2d 5861 . . . . . . 7  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  ( K `  x )  =  ( K `  <. X ,  Y >. ) )
32 df-ov 6278 . . . . . . 7  |-  ( X K Y )  =  ( K `  <. X ,  Y >. )
3331, 32syl6eqr 2519 . . . . . 6  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  ( K `  x )  =  ( X K Y ) )
3430, 33eleqtrrd 2551 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  F  e.  ( K `  x
) )
3534adantr 465 . . . 4  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  g  =  G )  ->  F  e.  ( K `
 x ) )
36 opex 4704 . . . . 5  |-  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.  .x.  ( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) ) >.  e.  _V
3736a1i 11 . . . 4  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.  .x.  ( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) ) >.  e.  _V )
3820fveq2d 5861 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  ( 1st `  x )  =  ( 1st `  <. X ,  Y >. )
)
39 op1stg 6786 . . . . . . . . . . . . 13  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 1st `  <. X ,  Y >. )  =  X )
408, 9, 39syl2anc 661 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1st `  <. X ,  Y >. )  =  X )
4140adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  ( 1st `  <. X ,  Y >. )  =  X )
4238, 41eqtrd 2501 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  ( 1st `  x )  =  X )
4342adantr 465 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( 1st `  x )  =  X )
4443fveq2d 5861 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( 1st `  ( 1st `  x
) )  =  ( 1st `  X ) )
4525adantr 465 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( 2nd `  x )  =  Y )
4645fveq2d 5861 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( 1st `  ( 2nd `  x
) )  =  ( 1st `  Y ) )
4744, 46opeq12d 4214 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.  =  <. ( 1st `  X
) ,  ( 1st `  Y ) >. )
48 simplrr 760 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  y  =  Z )
4948fveq2d 5861 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( 1st `  y )  =  ( 1st `  Z
) )
5047, 49oveq12d 6293 . . . . . 6  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( <. ( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>.  .x.  ( 1st `  y
) )  =  (
<. ( 1st `  X
) ,  ( 1st `  Y ) >.  .x.  ( 1st `  Z ) ) )
51 simprl 755 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  g  =  G )
5251fveq2d 5861 . . . . . 6  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( 1st `  g )  =  ( 1st `  G
) )
53 simprr 756 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  f  =  F )
5453fveq2d 5861 . . . . . 6  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( 1st `  f )  =  ( 1st `  F
) )
5550, 52, 54oveq123d 6296 . . . . 5  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.  .x.  ( 1st `  y
) ) ( 1st `  f ) )  =  ( ( 1st `  G
) ( <. ( 1st `  X ) ,  ( 1st `  Y
) >.  .x.  ( 1st `  Z ) ) ( 1st `  F ) ) )
5643fveq2d 5861 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( 2nd `  ( 1st `  x
) )  =  ( 2nd `  X ) )
5745fveq2d 5861 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( 2nd `  ( 2nd `  x
) )  =  ( 2nd `  Y ) )
5856, 57opeq12d 4214 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  =  <. ( 2nd `  X
) ,  ( 2nd `  Y ) >. )
5948fveq2d 5861 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( 2nd `  y )  =  ( 2nd `  Z
) )
6058, 59oveq12d 6293 . . . . . 6  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( <. ( 2nd `  ( 1st `  x ) ) ,  ( 2nd `  ( 2nd `  x ) )
>.  .xb  ( 2nd `  y
) )  =  (
<. ( 2nd `  X
) ,  ( 2nd `  Y ) >.  .xb  ( 2nd `  Z ) ) )
6151fveq2d 5861 . . . . . 6  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( 2nd `  g )  =  ( 2nd `  G
) )
6253fveq2d 5861 . . . . . 6  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( 2nd `  f )  =  ( 2nd `  F
) )
6360, 61, 62oveq123d 6296 . . . . 5  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  (
( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) )  =  ( ( 2nd `  G
) ( <. ( 2nd `  X ) ,  ( 2nd `  Y
) >.  .xb  ( 2nd `  Z
) ) ( 2nd `  F ) ) )
6455, 63opeq12d 4214 . . . 4  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  <. (
( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.  .x.  ( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) ) >.  =  <. ( ( 1st `  G ) ( <.
( 1st `  X
) ,  ( 1st `  Y ) >.  .x.  ( 1st `  Z ) ) ( 1st `  F
) ) ,  ( ( 2nd `  G
) ( <. ( 2nd `  X ) ,  ( 2nd `  Y
) >.  .xb  ( 2nd `  Z
) ) ( 2nd `  F ) ) >.
)
6528, 35, 37, 64ovmpt2dv2 6411 . . 3  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  (
( <. X ,  Y >. O Z )  =  ( g  e.  ( ( 2nd `  x
) K y ) ,  f  e.  ( K `  x ) 
|->  <. ( ( 1st `  g ) ( <.
( 1st `  ( 1st `  x ) ) ,  ( 1st `  ( 2nd `  x ) )
>.  .x.  ( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) ) >.
)  ->  ( G
( <. X ,  Y >. O Z ) F )  =  <. (
( 1st `  G
) ( <. ( 1st `  X ) ,  ( 1st `  Y
) >.  .x.  ( 1st `  Z ) ) ( 1st `  F ) ) ,  ( ( 2nd `  G ) ( <. ( 2nd `  X
) ,  ( 2nd `  Y ) >.  .xb  ( 2nd `  Z ) ) ( 2nd `  F
) ) >. )
)
6611, 13, 17, 65ovmpt2dv 6410 . 2  |-  ( ph  ->  ( O  =  ( x  e.  ( B  X.  B ) ,  y  e.  B  |->  ( g  e.  ( ( 2nd `  x ) K y ) ,  f  e.  ( K `
 x )  |->  <.
( ( 1st `  g
) ( <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.  .x.  ( 1st `  y
) ) ( 1st `  f ) ) ,  ( ( 2nd `  g
) ( <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  .xb  ( 2nd `  y
) ) ( 2nd `  f ) ) >.
) )  ->  ( G ( <. X ,  Y >. O Z ) F )  =  <. ( ( 1st `  G
) ( <. ( 1st `  X ) ,  ( 1st `  Y
) >.  .x.  ( 1st `  Z ) ) ( 1st `  F ) ) ,  ( ( 2nd `  G ) ( <. ( 2nd `  X
) ,  ( 2nd `  Y ) >.  .xb  ( 2nd `  Z ) ) ( 2nd `  F
) ) >. )
)
677, 66mpi 17 1  |-  ( ph  ->  ( G ( <. X ,  Y >. O Z ) F )  =  <. ( ( 1st `  G ) ( <.
( 1st `  X
) ,  ( 1st `  Y ) >.  .x.  ( 1st `  Z ) ) ( 1st `  F
) ) ,  ( ( 2nd `  G
) ( <. ( 2nd `  X ) ,  ( 2nd `  Y
) >.  .xb  ( 2nd `  Z
) ) ( 2nd `  F ) ) >.
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   _Vcvv 3106   <.cop 4026    X. cxp 4990   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277   1stc1st 6772   2ndc2nd 6773   Basecbs 14479   Hom chom 14555  compcco 14556    X.c cxpc 15284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-fz 11662  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-hom 14568  df-cco 14569  df-xpc 15288
This theorem is referenced by:  xpcco1st  15300  xpcco2nd  15301  xpcco2  15303  xpccatid  15304
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