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Theorem xpcco 15012
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 15011 . 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 4890 . . . 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 6135 . . . . 5  |-  ( ( 2nd `  x ) K y )  e. 
_V
15 fvex 5720 . . . . 5  |-  ( K `
 x )  e. 
_V
1614, 15mpt2ex 6669 . . . 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 5714 . . . . . . 7  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  ( 2nd `  x )  =  ( 2nd `  <. X ,  Y >. )
)
22 op2ndg 6609 . . . . . . . . 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 2475 . . . . . 6  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  ( 2nd `  x )  =  Y )
26 simprr 756 . . . . . 6  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  y  =  Z )
2725, 26oveq12d 6128 . . . . 5  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  (
( 2nd `  x
) K y )  =  ( Y K Z ) )
2819, 27eleqtrrd 2520 . . . 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 5714 . . . . . . 7  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  ( K `  x )  =  ( K `  <. X ,  Y >. ) )
32 df-ov 6113 . . . . . . 7  |-  ( X K Y )  =  ( K `  <. X ,  Y >. )
3331, 32syl6eqr 2493 . . . . . 6  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  ( K `  x )  =  ( X K Y ) )
3430, 33eleqtrrd 2520 . . . . 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 4575 . . . . 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 5714 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  =  <. X ,  Y >.  /\  y  =  Z ) )  ->  ( 1st `  x )  =  ( 1st `  <. X ,  Y >. )
)
39 op1stg 6608 . . . . . . . . . . . . 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 2475 . . . . . . . . . 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 5714 . . . . . . . 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 5714 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( 1st `  ( 2nd `  x
) )  =  ( 1st `  Y ) )
4744, 46opeq12d 4086 . . . . . . 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 5714 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( 1st `  y )  =  ( 1st `  Z
) )
5047, 49oveq12d 6128 . . . . . 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 5714 . . . . . 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 5714 . . . . . 6  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( 1st `  f )  =  ( 1st `  F
) )
5550, 52, 54oveq123d 6131 . . . . 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 5714 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( 2nd `  ( 1st `  x
) )  =  ( 2nd `  X ) )
5745fveq2d 5714 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( 2nd `  ( 2nd `  x
) )  =  ( 2nd `  Y ) )
5856, 57opeq12d 4086 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  <. ( 2nd `  ( 1st `  x
) ) ,  ( 2nd `  ( 2nd `  x ) ) >.  =  <. ( 2nd `  X
) ,  ( 2nd `  Y ) >. )
5948fveq2d 5714 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( 2nd `  y )  =  ( 2nd `  Z
) )
6058, 59oveq12d 6128 . . . . . 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 5714 . . . . . 6  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( 2nd `  g )  =  ( 2nd `  G
) )
6253fveq2d 5714 . . . . . 6  |-  ( ( ( ph  /\  (
x  =  <. X ,  Y >.  /\  y  =  Z ) )  /\  ( g  =  G  /\  f  =  F ) )  ->  ( 2nd `  f )  =  ( 2nd `  F
) )
6360, 61, 62oveq123d 6131 . . . . 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 4086 . . . 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 6243 . . 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 6242 . 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 1369    e. wcel 1756   _Vcvv 2991   <.cop 3902    X. cxp 4857   ` cfv 5437  (class class class)co 6110    e. cmpt2 6112   1stc1st 6594   2ndc2nd 6595   Basecbs 14193   Hom chom 14268  compcco 14269    X.c cxpc 14997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-int 4148  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-om 6496  df-1st 6596  df-2nd 6597  df-recs 6851  df-rdg 6885  df-1o 6939  df-oadd 6943  df-er 7120  df-en 7330  df-dom 7331  df-sdom 7332  df-fin 7333  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-nn 10342  df-2 10399  df-3 10400  df-4 10401  df-5 10402  df-6 10403  df-7 10404  df-8 10405  df-9 10406  df-10 10407  df-n0 10599  df-z 10666  df-dec 10775  df-uz 10881  df-fz 11457  df-struct 14195  df-ndx 14196  df-slot 14197  df-base 14198  df-hom 14281  df-cco 14282  df-xpc 15001
This theorem is referenced by:  xpcco1st  15013  xpcco2nd  15014  xpcco2  15016  xpccatid  15017
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