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Theorem 0catg 14956
Description: Any structure with an empty set of objects is a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
Assertion
Ref Expression
0catg  |-  ( ( C  e.  V  /\  (/)  =  ( Base `  C
) )  ->  C  e.  Cat )

Proof of Theorem 0catg
Dummy variables  f 
g  h  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 461 . 2  |-  ( ( C  e.  V  /\  (/)  =  ( Base `  C
) )  ->  (/)  =  (
Base `  C )
)
2 eqidd 2442 . 2  |-  ( ( C  e.  V  /\  (/)  =  ( Base `  C
) )  ->  ( Hom  `  C )  =  ( Hom  `  C
) )
3 eqidd 2442 . 2  |-  ( ( C  e.  V  /\  (/)  =  ( Base `  C
) )  ->  (comp `  C )  =  (comp `  C ) )
4 simpl 457 . 2  |-  ( ( C  e.  V  /\  (/)  =  ( Base `  C
) )  ->  C  e.  V )
5 noel 3771 . . . 4  |-  -.  x  e.  (/)
65pm2.21i 131 . . 3  |-  ( x  e.  (/)  ->  (/)  e.  ( x ( Hom  `  C
) x ) )
76adantl 466 . 2  |-  ( ( ( C  e.  V  /\  (/)  =  ( Base `  C ) )  /\  x  e.  (/) )  ->  (/) 
e.  ( x ( Hom  `  C )
x ) )
8 simpr1 1001 . . 3  |-  ( ( ( C  e.  V  /\  (/)  =  ( Base `  C ) )  /\  ( x  e.  (/)  /\  y  e.  (/)  /\  f  e.  ( y ( Hom  `  C ) x ) ) )  ->  x  e.  (/) )
95pm2.21i 131 . . 3  |-  ( x  e.  (/)  ->  ( (/) ( <.
y ,  x >. (comp `  C ) x ) f )  =  f )
108, 9syl 16 . 2  |-  ( ( ( C  e.  V  /\  (/)  =  ( Base `  C ) )  /\  ( x  e.  (/)  /\  y  e.  (/)  /\  f  e.  ( y ( Hom  `  C ) x ) ) )  ->  ( (/) ( <. y ,  x >. (comp `  C )
x ) f )  =  f )
11 simpr1 1001 . . 3  |-  ( ( ( C  e.  V  /\  (/)  =  ( Base `  C ) )  /\  ( x  e.  (/)  /\  y  e.  (/)  /\  f  e.  ( x ( Hom  `  C ) y ) ) )  ->  x  e.  (/) )
125pm2.21i 131 . . 3  |-  ( x  e.  (/)  ->  ( f
( <. x ,  x >. (comp `  C )
y ) (/) )  =  f )
1311, 12syl 16 . 2  |-  ( ( ( C  e.  V  /\  (/)  =  ( Base `  C ) )  /\  ( x  e.  (/)  /\  y  e.  (/)  /\  f  e.  ( x ( Hom  `  C ) y ) ) )  ->  (
f ( <. x ,  x >. (comp `  C
) y ) (/) )  =  f )
14 simp21 1028 . . 3  |-  ( ( ( C  e.  V  /\  (/)  =  ( Base `  C ) )  /\  ( x  e.  (/)  /\  y  e.  (/)  /\  z  e.  (/) )  /\  (
f  e.  ( x ( Hom  `  C
) y )  /\  g  e.  ( y
( Hom  `  C ) z ) ) )  ->  x  e.  (/) )
155pm2.21i 131 . . 3  |-  ( x  e.  (/)  ->  ( g
( <. x ,  y
>. (comp `  C )
z ) f )  e.  ( x ( Hom  `  C )
z ) )
1614, 15syl 16 . 2  |-  ( ( ( C  e.  V  /\  (/)  =  ( Base `  C ) )  /\  ( x  e.  (/)  /\  y  e.  (/)  /\  z  e.  (/) )  /\  (
f  e.  ( x ( Hom  `  C
) y )  /\  g  e.  ( y
( Hom  `  C ) z ) ) )  ->  ( g (
<. x ,  y >.
(comp `  C )
z ) f )  e.  ( x ( Hom  `  C )
z ) )
17 simp2ll 1062 . . 3  |-  ( ( ( C  e.  V  /\  (/)  =  ( Base `  C ) )  /\  ( ( x  e.  (/)  /\  y  e.  (/) )  /\  ( z  e.  (/)  /\  w  e.  (/) ) )  /\  (
f  e.  ( x ( Hom  `  C
) y )  /\  g  e.  ( y
( Hom  `  C ) z )  /\  h  e.  ( z ( Hom  `  C ) w ) ) )  ->  x  e.  (/) )
185pm2.21i 131 . . 3  |-  ( x  e.  (/)  ->  ( (
h ( <. y ,  z >. (comp `  C ) w ) g ) ( <.
x ,  y >.
(comp `  C )
w ) f )  =  ( h (
<. x ,  z >.
(comp `  C )
w ) ( g ( <. x ,  y
>. (comp `  C )
z ) f ) ) )
1917, 18syl 16 . 2  |-  ( ( ( C  e.  V  /\  (/)  =  ( Base `  C ) )  /\  ( ( x  e.  (/)  /\  y  e.  (/) )  /\  ( z  e.  (/)  /\  w  e.  (/) ) )  /\  (
f  e.  ( x ( Hom  `  C
) y )  /\  g  e.  ( y
( Hom  `  C ) z )  /\  h  e.  ( z ( Hom  `  C ) w ) ) )  ->  (
( h ( <.
y ,  z >.
(comp `  C )
w ) g ) ( <. x ,  y
>. (comp `  C )
w ) f )  =  ( h (
<. x ,  z >.
(comp `  C )
w ) ( g ( <. x ,  y
>. (comp `  C )
z ) f ) ) )
201, 2, 3, 4, 7, 10, 13, 16, 19iscatd 14942 1  |-  ( ( C  e.  V  /\  (/)  =  ( Base `  C
) )  ->  C  e.  Cat )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802   (/)c0 3767   <.cop 4016   ` cfv 5574  (class class class)co 6277   Basecbs 14504   Hom chom 14580  compcco 14581   Catccat 14933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-nul 4562
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-iota 5537  df-fv 5582  df-ov 6280  df-cat 14937
This theorem is referenced by:  0cat  14957
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