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Theorem 0catg 15536
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 462 . 2  |-  ( ( C  e.  V  /\  (/)  =  ( Base `  C
) )  ->  (/)  =  (
Base `  C )
)
2 eqidd 2429 . 2  |-  ( ( C  e.  V  /\  (/)  =  ( Base `  C
) )  ->  ( Hom  `  C )  =  ( Hom  `  C
) )
3 eqidd 2429 . 2  |-  ( ( C  e.  V  /\  (/)  =  ( Base `  C
) )  ->  (comp `  C )  =  (comp `  C ) )
4 simpl 458 . 2  |-  ( ( C  e.  V  /\  (/)  =  ( Base `  C
) )  ->  C  e.  V )
5 noel 3708 . . . 4  |-  -.  x  e.  (/)
65pm2.21i 134 . . 3  |-  ( x  e.  (/)  ->  (/)  e.  ( x ( Hom  `  C
) x ) )
76adantl 467 . 2  |-  ( ( ( C  e.  V  /\  (/)  =  ( Base `  C ) )  /\  x  e.  (/) )  ->  (/) 
e.  ( x ( Hom  `  C )
x ) )
8 simpr1 1011 . . 3  |-  ( ( ( C  e.  V  /\  (/)  =  ( Base `  C ) )  /\  ( x  e.  (/)  /\  y  e.  (/)  /\  f  e.  ( y ( Hom  `  C ) x ) ) )  ->  x  e.  (/) )
95pm2.21i 134 . . 3  |-  ( x  e.  (/)  ->  ( (/) ( <.
y ,  x >. (comp `  C ) x ) f )  =  f )
108, 9syl 17 . 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 1011 . . 3  |-  ( ( ( C  e.  V  /\  (/)  =  ( Base `  C ) )  /\  ( x  e.  (/)  /\  y  e.  (/)  /\  f  e.  ( x ( Hom  `  C ) y ) ) )  ->  x  e.  (/) )
125pm2.21i 134 . . 3  |-  ( x  e.  (/)  ->  ( f
( <. x ,  x >. (comp `  C )
y ) (/) )  =  f )
1311, 12syl 17 . 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 1038 . . 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 134 . . 3  |-  ( x  e.  (/)  ->  ( g
( <. x ,  y
>. (comp `  C )
z ) f )  e.  ( x ( Hom  `  C )
z ) )
1614, 15syl 17 . 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 1072 . . 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 134 . . 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 17 . 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 15522 1  |-  ( ( C  e.  V  /\  (/)  =  ( Base `  C
) )  ->  C  e.  Cat )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   (/)c0 3704   <.cop 3947   ` cfv 5544  (class class class)co 6249   Basecbs 15064   Hom chom 15144  compcco 15145   Catccat 15513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-nul 4498
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-iota 5508  df-fv 5552  df-ov 6252  df-cat 15517
This theorem is referenced by:  0cat  15537
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