Theorem cidfn 16340

Description: The identity arrow operator is a function from objects to arrows. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
cidfn.b	|- B = ( Base ` C )
cidfn.i	|- .1. = ( Id ` C )

Assertion

Ref	Expression
cidfn	|- ( C e. Cat -> .1. Fn B )

Proof of Theorem cidfn

Dummy variables f g x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		riotaex 6615	. . 3 |- ( iota_ g e. ( x ( Hom ` C ) x ) A. y e. B ( A. f e. ( y ( Hom ` C ) x ) ( g ( <. y , x >. (comp ` C ) x ) f ) = f /\ A. f e. ( x ( Hom ` C ) y ) ( f ( <. x , x >. (comp ` C ) y ) g ) = f ) ) e. _V
2		eqid 2622	. . 3 |- ( x e. B |-> ( iota_ g e. ( x ( Hom ` C ) x ) A. y e. B ( A. f e. ( y ( Hom ` C ) x ) ( g ( <. y , x >. (comp ` C ) x ) f ) = f /\ A. f e. ( x ( Hom ` C ) y ) ( f ( <. x , x >. (comp ` C ) y ) g ) = f ) ) ) = ( x e. B |-> ( iota_ g e. ( x ( Hom ` C ) x ) A. y e. B ( A. f e. ( y ( Hom ` C ) x ) ( g ( <. y , x >. (comp ` C ) x ) f ) = f /\ A. f e. ( x ( Hom ` C ) y ) ( f ( <. x , x >. (comp ` C ) y ) g ) = f ) ) )
3	1, 2	fnmpti 6022	. 2 |- ( x e. B |-> ( iota_ g e. ( x ( Hom ` C ) x ) A. y e. B ( A. f e. ( y ( Hom ` C ) x ) ( g ( <. y , x >. (comp ` C ) x ) f ) = f /\ A. f e. ( x ( Hom ` C ) y ) ( f ( <. x , x >. (comp ` C ) y ) g ) = f ) ) ) Fn B
4		cidfn.b	. . . 4 |- B = ( Base ` C )
5		eqid 2622	. . . 4 |- ( Hom ` C ) = ( Hom ` C )
6		eqid 2622	. . . 4 |- (comp ` C ) = (comp ` C )
7		id 22	. . . 4 |- ( C e. Cat -> C e. Cat )
8		cidfn.i	. . . 4 |- .1. = ( Id ` C )
9	4, 5, 6, 7, 8	cidfval 16337	. . 3 |- ( C e. Cat -> .1. = ( x e. B |-> ( iota_ g e. ( x ( Hom ` C ) x ) A. y e. B ( A. f e. ( y ( Hom ` C ) x ) ( g ( <. y , x >. (comp ` C ) x ) f ) = f /\ A. f e. ( x ( Hom ` C ) y ) ( f ( <. x , x >. (comp ` C ) y ) g ) = f ) ) ) )
10	9	fneq1d 5981	. 2 |- ( C e. Cat -> ( .1. Fn B <-> ( x e. B |-> ( iota_ g e. ( x ( Hom ` C ) x ) A. y e. B ( A. f e. ( y ( Hom ` C ) x ) ( g ( <. y , x >. (comp ` C ) x ) f ) = f /\ A. f e. ( x ( Hom ` C ) y ) ( f ( <. x , x >. (comp ` C ) y ) g ) = f ) ) ) Fn B ) )
11	3, 10	mpbiri 248	1 |- ( C e. Cat -> .1. Fn B )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   <.cop 4183   |-> cmpt 4729   Fn wfn 5883   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Idccid 16326

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-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-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  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-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-riota 6611  df-ov 6653  df-cid 16330

This theorem is referenced by:  oppccatid  16379  fucidcl  16625  fucsect  16632  curfcl  16872  curf2ndf  16887