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Theorem 0subcat 15736
Description: For any category  C, the empty set is a (full) subcategory of  C, see example 4.3(1.a) in [Adamek] p. 48. (Contributed by AV, 23-Apr-2020.)
Assertion
Ref Expression
0subcat  |-  ( C  e.  Cat  ->  (/)  e.  (Subcat `  C ) )

Proof of Theorem 0subcat
Dummy variables  f 
g  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ssc 15735 . 2  |-  ( C  e.  Cat  ->  (/)  C_cat  ( Hom f  `  C ) )
2 ral0 3903 . . 3  |-  A. x  e.  (/)  ( ( ( Id `  C ) `
 x )  e.  ( x (/) x )  /\  A. y  e.  (/)  A. z  e.  (/)  A. f  e.  ( x
(/) y ) A. g  e.  ( y (/) z ) ( g ( <. x ,  y
>. (comp `  C )
z ) f )  e.  ( x (/) z ) )
32a1i 11 . 2  |-  ( C  e.  Cat  ->  A. x  e.  (/)  ( ( ( Id `  C ) `
 x )  e.  ( x (/) x )  /\  A. y  e.  (/)  A. z  e.  (/)  A. f  e.  ( x
(/) y ) A. g  e.  ( y (/) z ) ( g ( <. x ,  y
>. (comp `  C )
z ) f )  e.  ( x (/) z ) ) )
4 eqid 2423 . . 3  |-  ( Hom f  `  C )  =  ( Hom f  `  C )
5 eqid 2423 . . 3  |-  ( Id
`  C )  =  ( Id `  C
)
6 eqid 2423 . . 3  |-  (comp `  C )  =  (comp `  C )
7 id 23 . . 3  |-  ( C  e.  Cat  ->  C  e.  Cat )
8 f0 5779 . . . . . 6  |-  (/) : (/) --> (/)
9 ffn 5744 . . . . . 6  |-  ( (/) :
(/) --> (/)  ->  (/)  Fn  (/) )
108, 9ax-mp 5 . . . . 5  |-  (/)  Fn  (/)
11 0xp 4932 . . . . . 6  |-  ( (/)  X.  (/) )  =  (/)
1211fneq2i 5687 . . . . 5  |-  ( (/)  Fn  ( (/)  X.  (/) )  <->  (/)  Fn  (/) )
1310, 12mpbir 213 . . . 4  |-  (/)  Fn  ( (/) 
X.  (/) )
1413a1i 11 . . 3  |-  ( C  e.  Cat  ->  (/)  Fn  ( (/) 
X.  (/) ) )
154, 5, 6, 7, 14issubc2 15734 . 2  |-  ( C  e.  Cat  ->  ( (/) 
e.  (Subcat `  C )  <->  (
(/)  C_cat  ( Hom f  `  C )  /\  A. x  e.  (/)  ( ( ( Id `  C
) `  x )  e.  ( x (/) x )  /\  A. y  e.  (/)  A. z  e.  (/)  A. f  e.  ( x
(/) y ) A. g  e.  ( y (/) z ) ( g ( <. x ,  y
>. (comp `  C )
z ) f )  e.  ( x (/) z ) ) ) ) )
161, 3, 15mpbir2and 931 1  |-  ( C  e.  Cat  ->  (/)  e.  (Subcat `  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    e. wcel 1869   A.wral 2776   (/)c0 3762   <.cop 4003   class class class wbr 4421    X. cxp 4849    Fn wfn 5594   -->wf 5595   ` cfv 5599  (class class class)co 6303  compcco 15195   Catccat 15563   Idccid 15564   Hom f chomf 15565    C_cat cssc 15705  Subcatcsubc 15707
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-fal 1444  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-1st 6805  df-2nd 6806  df-pm 7481  df-ixp 7529  df-homf 15569  df-ssc 15708  df-subc 15710
This theorem is referenced by: (None)
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