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Theorem xpcco 15776
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 15775 . 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 4855 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
118, 9, 10syl2anc 659 . . 3  |-  ( ph  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
12 xpcco.z . . . 4  |-  ( ph  ->  Z  e.  B )
1312adantr 463 . . 3  |-  ( (
ph  /\  x  =  <. X ,  Y >. )  ->  Z  e.  B
)
14 ovex 6306 . . . . 5  |-  ( ( 2nd `  x ) K y )  e. 
_V
15 fvex 5859 . . . . 5  |-  ( K `
 x )  e. 
_V
1614, 15mpt2ex 6861 . . . 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 463 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  G  e.  ( Y K Z ) )
20 simprl 756 . . . . . . . 8  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  x  =  <. X ,  Y >. )
2120fveq2d 5853 . . . . . . 7  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  ( 2nd `  x )  =  ( 2nd `  <. X ,  Y >. )
)
22 op2ndg 6797 . . . . . . . . 9  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
238, 9, 22syl2anc 659 . . . . . . . 8  |-  ( ph  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
2423adantr 463 . . . . . . 7  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  ( 2nd `  <. X ,  Y >. )  =  Y )
2521, 24eqtrd 2443 . . . . . 6  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  ( 2nd `  x )  =  Y )
26 simprr 758 . . . . . 6  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  y  =  Z )
2725, 26oveq12d 6296 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  (
( 2nd `  x
) K y )  =  ( Y K Z ) )
2819, 27eleqtrrd 2493 . . . 4  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  G  e.  ( ( 2nd `  x
) K y ) )
29 xpcco.f . . . . . . 7  |-  ( ph  ->  F  e.  ( X K Y ) )
3029adantr 463 . . . . . 6  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  F  e.  ( X K Y ) )
3120fveq2d 5853 . . . . . . 7  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  ( K `  x )  =  ( K `  <. X ,  Y >. ) )
32 df-ov 6281 . . . . . . 7  |-  ( X K Y )  =  ( K `  <. X ,  Y >. )
3331, 32syl6eqr 2461 . . . . . 6  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  ( K `  x )  =  ( X K Y ) )
3430, 33eleqtrrd 2493 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  F  e.  ( K `  x
) )
3534adantr 463 . . . 4  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  g  =  G )  ->  F  e.  ( K `
 x ) )
36 opex 4655 . . . . 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 5853 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  ( 1st `  x )  =  ( 1st `  <. X ,  Y >. )
)
39 op1stg 6796 . . . . . . . . . . . . 13  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( 1st `  <. X ,  Y >. )  =  X )
408, 9, 39syl2anc 659 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1st `  <. X ,  Y >. )  =  X )
4140adantr 463 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  ( 1st `  <. X ,  Y >. )  =  X )
4238, 41eqtrd 2443 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  ( 1st `  x )  =  X )
4342adantr 463 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( 1st `  x )  =  X )
4443fveq2d 5853 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( 1st `  ( 1st `  x
) )  =  ( 1st `  X ) )
4525adantr 463 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( 2nd `  x )  =  Y )
4645fveq2d 5853 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( 1st `  ( 2nd `  x
) )  =  ( 1st `  Y ) )
4744, 46opeq12d 4167 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  <. ( 1st `  ( 1st `  x
) ) ,  ( 1st `  ( 2nd `  x ) ) >.  =  <. ( 1st `  X
) ,  ( 1st `  Y ) >. )
48 simplrr 763 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  y  =  Z )
4948fveq2d 5853 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( 1st `  y )  =  ( 1st `  Z
) )
5047, 49oveq12d 6296 . . . . . 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 756 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  g  =  G )
5251fveq2d 5853 . . . . . 6  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( 1st `  g )  =  ( 1st `  G
) )
53 simprr 758 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  f  =  F )
5453fveq2d 5853 . . . . . 6  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( 1st `  f )  =  ( 1st `  F
) )
5550, 52, 54oveq123d 6299 . . . . 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 5853 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( 2nd `  ( 1st `  x
) )  =  ( 2nd `  X ) )
5745fveq2d 5853 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( 2nd `  ( 2nd `  x
) )  =  ( 2nd `  Y ) )
5856, 57opeq12d 4167 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  =  <. ( 2nd `  X
) ,  ( 2nd `  Y ) >. )
5948fveq2d 5853 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( 2nd `  y )  =  ( 2nd `  Z
) )
6058, 59oveq12d 6296 . . . . . 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 5853 . . . . . 6  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( 2nd `  g )  =  ( 2nd `  G
) )
6253fveq2d 5853 . . . . . 6  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( 2nd `  f )  =  ( 2nd `  F
) )
6360, 61, 62oveq123d 6299 . . . . 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 4167 . . . 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 6417 . . 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 6416 . 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 20 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 367    = wceq 1405    e. wcel 1842   _Vcvv 3059   <.cop 3978    X. cxp 4821   ` cfv 5569  (class class class)co 6278    |-> cmpt2 6280   1stc1st 6782   2ndc2nd 6783   Basecbs 14841   Hom chom 14920  compcco 14921    X.c cxpc 15761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-fz 11727  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-hom 14933  df-cco 14934  df-xpc 15765
This theorem is referenced by:  xpcco1st  15777  xpcco2nd  15778  xpcco2  15780  xpccatid  15781
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