Theorem fuchom 16621

Description: The morphisms in the functor category are natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
fucbas.q	|- Q = ( C FuncCat D )
fuchom.n	|- N = ( C Nat D )

Assertion

Ref	Expression
fuchom	|- N = ( Hom ` Q )

Proof of Theorem fuchom

Dummy variables a b f g h v x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fucbas.q	. . . . 5 |- Q = ( C FuncCat D )
2		eqid 2622	. . . . 5 |- ( C Func D ) = ( C Func D )
3		fuchom.n	. . . . 5 |- N = ( C Nat D )
4		eqid 2622	. . . . 5 |- ( Base ` C ) = ( Base ` C )
5		eqid 2622	. . . . 5 |- (comp ` D ) = (comp ` D )
6		simpl 473	. . . . 5 |- ( ( C e. Cat /\ D e. Cat ) -> C e. Cat )
7		simpr 477	. . . . 5 |- ( ( C e. Cat /\ D e. Cat ) -> D e. Cat )
8		eqid 2622	. . . . . 6 |- (comp ` Q ) = (comp ` Q )
9	1, 2, 3, 4, 5, 6, 7, 8	fuccofval 16619	. . . . 5 |- ( ( C e. Cat /\ D e. Cat ) -> (comp ` Q ) = ( v e. ( ( C Func D ) X. ( C Func D ) ) , h e. ( C Func D ) |-> [_ ( 1st ` v ) / f ]_ [_ ( 2nd ` v ) / g ]_ ( b e. ( g N h ) , a e. ( f N g ) |-> ( x e. ( Base ` C ) |-> ( ( b ` x ) ( <. ( ( 1st ` f ) ` x ) , ( ( 1st ` g ) ` x ) >. (comp ` D ) ( ( 1st ` h ) ` x ) ) ( a ` x ) ) ) ) ) )
10	1, 2, 3, 4, 5, 6, 7, 9	fucval 16618	. . . 4 |- ( ( C e. Cat /\ D e. Cat ) -> Q = { <. ( Base ` ndx ) , ( C Func D ) >. , <. ( Hom ` ndx ) , N >. , <. (comp ` ndx ) , (comp ` Q ) >. } )
11		catstr 16617	. . . 4 |- { <. ( Base ` ndx ) , ( C Func D ) >. , <. ( Hom ` ndx ) , N >. , <. (comp ` ndx ) , (comp ` Q ) >. } Struct <. 1 , ; 1 5 >.
12		homid 16075	. . . 4 |- Hom = Slot ( Hom ` ndx )
13		snsstp2 4348	. . . 4 |- { <. ( Hom ` ndx ) , N >. } C_ { <. ( Base ` ndx ) , ( C Func D ) >. , <. ( Hom ` ndx ) , N >. , <. (comp ` ndx ) , (comp ` Q ) >. }
14		ovex 6678	. . . . . 6 |- ( C Nat D ) e. _V
15	3, 14	eqeltri 2697	. . . . 5 |- N e. _V
16	15	a1i 11	. . . 4 |- ( ( C e. Cat /\ D e. Cat ) -> N e. _V )
17		eqid 2622	. . . 4 |- ( Hom ` Q ) = ( Hom ` Q )
18	10, 11, 12, 13, 16, 17	strfv3 15908	. . 3 |- ( ( C e. Cat /\ D e. Cat ) -> ( Hom ` Q ) = N )
19	18	eqcomd 2628	. 2 |- ( ( C e. Cat /\ D e. Cat ) -> N = ( Hom ` Q ) )
20		df-hom 15966	. . . 4 |- Hom = Slot ; 1 4
21	20	str0 15911	. . 3 |- (/) = ( Hom ` (/) )
22	3	natffn 16609	. . . . 5 |- N Fn ( ( C Func D ) X. ( C Func D ) )
23		funcrcl 16523	. . . . . . . . . 10 |- ( f e. ( C Func D ) -> ( C e. Cat /\ D e. Cat ) )
24	23	con3i 150	. . . . . . . . 9 |- ( -. ( C e. Cat /\ D e. Cat ) -> -. f e. ( C Func D ) )
25	24	eq0rdv 3979	. . . . . . . 8 |- ( -. ( C e. Cat /\ D e. Cat ) -> ( C Func D ) = (/) )
26	25	xpeq2d 5139	. . . . . . 7 |- ( -. ( C e. Cat /\ D e. Cat ) -> ( ( C Func D ) X. ( C Func D ) ) = ( ( C Func D ) X. (/) ) )
27		xp0 5552	. . . . . . 7 |- ( ( C Func D ) X. (/) ) = (/)
28	26, 27	syl6eq 2672	. . . . . 6 |- ( -. ( C e. Cat /\ D e. Cat ) -> ( ( C Func D ) X. ( C Func D ) ) = (/) )
29	28	fneq2d 5982	. . . . 5 |- ( -. ( C e. Cat /\ D e. Cat ) -> ( N Fn ( ( C Func D ) X. ( C Func D ) ) <-> N Fn (/) ) )
30	22, 29	mpbii 223	. . . 4 |- ( -. ( C e. Cat /\ D e. Cat ) -> N Fn (/) )
31		fn0 6011	. . . 4 |- ( N Fn (/) <-> N = (/) )
32	30, 31	sylib 208	. . 3 |- ( -. ( C e. Cat /\ D e. Cat ) -> N = (/) )
33		fnfuc 16605	. . . . . . 7 |- FuncCat Fn ( Cat X. Cat )
34		fndm 5990	. . . . . . 7 |- ( FuncCat Fn ( Cat X. Cat ) -> dom FuncCat = ( Cat X. Cat ) )
35	33, 34	ax-mp 5	. . . . . 6 |- dom FuncCat = ( Cat X. Cat )
36	35	ndmov 6818	. . . . 5 |- ( -. ( C e. Cat /\ D e. Cat ) -> ( C FuncCat D ) = (/) )
37	1, 36	syl5eq 2668	. . . 4 |- ( -. ( C e. Cat /\ D e. Cat ) -> Q = (/) )
38	37	fveq2d 6195	. . 3 |- ( -. ( C e. Cat /\ D e. Cat ) -> ( Hom ` Q ) = ( Hom ` (/) ) )
39	21, 32, 38	3eqtr4a 2682	. 2 |- ( -. ( C e. Cat /\ D e. Cat ) -> N = ( Hom ` Q ) )
40	19, 39	pm2.61i 176	1 |- N = ( Hom ` Q )

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   {ctp 4181   <.cop 4183   X. cxp 5112   dom cdm 5114   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   1c1 9937   4c4 11072   5c5 11073  ;cdc 11493   ndxcnx 15854   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Func cfunc 16514   Nat cnat 16601   FuncCat cfuc 16602

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-hom 15966  df-cco 15967  df-func 16518  df-nat 16603  df-fuc 16604

This theorem is referenced by:  fuccatid  16629  fucsect  16632  fuciso  16635  evlfcllem  16861  evlfcl  16862  curfcl  16872  uncf2  16877  curfuncf  16878  diag2cl  16886  curf2ndf  16887  yonedalem21  16913  yonedalem22  16918  yonedalem3b  16919  yonedalem3  16920  yonffthlem  16922