Theorem 0subcat 16498

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.

Step	Hyp	Ref	Expression
1		0ssc 16497	. 2 |- ( C e. Cat -> (/) C_cat ( Hom f ` C ) )
2		ral0 4076	. . 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 ) )
3	2	a1i 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 2622	. . 3 |- ( Hom f ` C ) = ( Hom f ` C )
5		eqid 2622	. . 3 |- ( Id ` C ) = ( Id ` C )
6		eqid 2622	. . 3 |- (comp ` C ) = (comp ` C )
7		id 22	. . 3 |- ( C e. Cat -> C e. Cat )
8		f0 6086	. . . . . 6 |- (/) : (/) --> (/)
9		ffn 6045	. . . . . 6 |- ( (/) : (/) --> (/) -> (/) Fn (/) )
10	8, 9	ax-mp 5	. . . . 5 |- (/) Fn (/)
11		0xp 5199	. . . . . 6 |- ( (/) X. (/) ) = (/)
12	11	fneq2i 5986	. . . . 5 |- ( (/) Fn ( (/) X. (/) ) <-> (/) Fn (/) )
13	10, 12	mpbir 221	. . . 4 |- (/) Fn ( (/) X. (/) )
14	13	a1i 11	. . 3 |- ( C e. Cat -> (/) Fn ( (/) X. (/) ) )
15	4, 5, 6, 7, 14	issubc2 16496	. 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 ) ) ) ) )
16	1, 3, 15	mpbir2and 957	1 |- ( C e. Cat -> (/) e. (Subcat ` C ) )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   A.wral 2912   (/)c0 3915   <.cop 4183   class class class wbr 4653   X. cxp 5112   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650  compcco 15953   Catccat 16325   Idccid 16326   Hom _f chomf 16327   C__cat cssc 16467  Subcatcsubc 16469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-pm 7860  df-ixp 7909  df-homf 16331  df-ssc 16470  df-subc 16472

This theorem is referenced by: (None)