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Theorem 0catg 13605
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 447 . 2  |-  ( ( C  e.  V  /\  (/)  =  ( Base `  C
) )  ->  (/)  =  (
Base `  C )
)
2 eqidd 2297 . 2  |-  ( ( C  e.  V  /\  (/)  =  ( Base `  C
) )  ->  (  Hom  `  C )  =  (  Hom  `  C
) )
3 eqidd 2297 . 2  |-  ( ( C  e.  V  /\  (/)  =  ( Base `  C
) )  ->  (comp `  C )  =  (comp `  C ) )
4 simpl 443 . 2  |-  ( ( C  e.  V  /\  (/)  =  ( Base `  C
) )  ->  C  e.  V )
5 noel 3472 . . . 4  |-  -.  x  e.  (/)
65pm2.21i 123 . . 3  |-  ( x  e.  (/)  ->  (/)  e.  ( x (  Hom  `  C
) x ) )
76adantl 452 . 2  |-  ( ( ( C  e.  V  /\  (/)  =  ( Base `  C ) )  /\  x  e.  (/) )  ->  (/) 
e.  ( x (  Hom  `  C )
x ) )
8 simpr1 961 . . 3  |-  ( ( ( C  e.  V  /\  (/)  =  ( Base `  C ) )  /\  ( x  e.  (/)  /\  y  e.  (/)  /\  f  e.  ( y (  Hom  `  C ) x ) ) )  ->  x  e.  (/) )
95pm2.21i 123 . . 3  |-  ( x  e.  (/)  ->  ( (/) ( <.
y ,  x >. (comp `  C ) x ) f )  =  f )
108, 9syl 15 . 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 961 . . 3  |-  ( ( ( C  e.  V  /\  (/)  =  ( Base `  C ) )  /\  ( x  e.  (/)  /\  y  e.  (/)  /\  f  e.  ( x (  Hom  `  C ) y ) ) )  ->  x  e.  (/) )
125pm2.21i 123 . . 3  |-  ( x  e.  (/)  ->  ( f
( <. x ,  x >. (comp `  C )
y ) (/) )  =  f )
1311, 12syl 15 . 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 988 . . 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 123 . . 3  |-  ( x  e.  (/)  ->  ( g
( <. x ,  y
>. (comp `  C )
z ) f )  e.  ( x (  Hom  `  C )
z ) )
1614, 15syl 15 . 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 1022 . . 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 123 . . 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 15 . 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 13591 1  |-  ( ( C  e.  V  /\  (/)  =  ( Base `  C
) )  ->  C  e.  Cat )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   (/)c0 3468   <.cop 3656   ` cfv 5271  (class class class)co 5874   Basecbs 13164    Hom chom 13235  compcco 13236   Catccat 13582
This theorem is referenced by:  0cat  13606
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-cat 13586
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