Theorem fuclid 15210

Description: Left identity of natural transformations. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
fuclid.q	|- Q = ( C FuncCat D )
fuclid.n	|- N = ( C Nat D )
fuclid.x	|- .xb = (comp ` Q )
fuclid.1	|- .1. = ( Id ` D )
fuclid.r	|- ( ph -> R e. ( F N G ) )

Assertion

Ref	Expression
fuclid	|- ( ph -> ( ( .1. o. ( 1st ` G ) ) ( <. F , G >. .xb G ) R ) = R )

Proof of Theorem fuclid

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2467	. . . . . . 7 |- ( Base ` C ) = ( Base ` C )
2		eqid 2467	. . . . . . 7 |- ( Base ` D ) = ( Base ` D )
3		relfunc 15106	. . . . . . . 8 |- Rel ( C Func D )
4		fuclid.r	. . . . . . . . . 10 |- ( ph -> R e. ( F N G ) )
5		fuclid.n	. . . . . . . . . . 11 |- N = ( C Nat D )
6	5	natrcl 15194	. . . . . . . . . 10 |- ( R e. ( F N G ) -> ( F e. ( C Func D ) /\ G e. ( C Func D ) ) )
7	4, 6	syl 16	. . . . . . . . 9 |- ( ph -> ( F e. ( C Func D ) /\ G e. ( C Func D ) ) )
8	7	simprd 463	. . . . . . . 8 |- ( ph -> G e. ( C Func D ) )
9		1st2ndbr 6844	. . . . . . . 8 |- ( ( Rel ( C Func D ) /\ G e. ( C Func D ) ) -> ( 1st ` G ) ( C Func D ) ( 2nd ` G ) )
10	3, 8, 9	sylancr 663	. . . . . . 7 |- ( ph -> ( 1st ` G ) ( C Func D ) ( 2nd ` G ) )
11	1, 2, 10	funcf1 15110	. . . . . 6 |- ( ph -> ( 1st ` G ) : ( Base ` C ) --> ( Base ` D ) )
12		fvco3 5951	. . . . . 6 |- ( ( ( 1st ` G ) : ( Base ` C ) --> ( Base ` D ) /\ x e. ( Base ` C ) ) -> ( ( .1. o. ( 1st ` G ) ) ` x ) = ( .1. ` ( ( 1st ` G ) ` x ) ) )
13	11, 12	sylan 471	. . . . 5 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( .1. o. ( 1st ` G ) ) ` x ) = ( .1. ` ( ( 1st ` G ) ` x ) ) )
14	13	oveq1d 6310	. . . 4 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( ( .1. o. ( 1st ` G ) ) ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. (comp ` D ) ( ( 1st ` G ) ` x ) ) ( R ` x ) ) = ( ( .1. ` ( ( 1st ` G ) ` x ) ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. (comp ` D ) ( ( 1st ` G ) ` x ) ) ( R ` x ) ) )
15		eqid 2467	. . . . 5 |- ( Hom ` D ) = ( Hom ` D )
16		fuclid.1	. . . . 5 |- .1. = ( Id ` D )
17	7	simpld 459	. . . . . . . 8 |- ( ph -> F e. ( C Func D ) )
18		funcrcl 15107	. . . . . . . 8 |- ( F e. ( C Func D ) -> ( C e. Cat /\ D e. Cat ) )
19	17, 18	syl 16	. . . . . . 7 |- ( ph -> ( C e. Cat /\ D e. Cat ) )
20	19	simprd 463	. . . . . 6 |- ( ph -> D e. Cat )
21	20	adantr 465	. . . . 5 |- ( ( ph /\ x e. ( Base ` C ) ) -> D e. Cat )
22		1st2ndbr 6844	. . . . . . . 8 |- ( ( Rel ( C Func D ) /\ F e. ( C Func D ) ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
23	3, 17, 22	sylancr 663	. . . . . . 7 |- ( ph -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
24	1, 2, 23	funcf1 15110	. . . . . 6 |- ( ph -> ( 1st ` F ) : ( Base ` C ) --> ( Base ` D ) )
25	24	ffvelrnda 6032	. . . . 5 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` F ) ` x ) e. ( Base ` D ) )
26		eqid 2467	. . . . 5 |- (comp ` D ) = (comp ` D )
27	11	ffvelrnda 6032	. . . . 5 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` G ) ` x ) e. ( Base ` D ) )
28	5, 4	nat1st2nd 15195	. . . . . . 7 |- ( ph -> R e. ( <. ( 1st ` F ) , ( 2nd ` F ) >. N <. ( 1st ` G ) , ( 2nd ` G ) >. ) )
29	28	adantr 465	. . . . . 6 |- ( ( ph /\ x e. ( Base ` C ) ) -> R e. ( <. ( 1st ` F ) , ( 2nd ` F ) >. N <. ( 1st ` G ) , ( 2nd ` G ) >. ) )
30		simpr 461	. . . . . 6 |- ( ( ph /\ x e. ( Base ` C ) ) -> x e. ( Base ` C ) )
31	5, 29, 1, 15, 30	natcl 15197	. . . . 5 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( R ` x ) e. ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` G ) ` x ) ) )
32	2, 15, 16, 21, 25, 26, 27, 31	catlid 14955	. . . 4 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( .1. ` ( ( 1st ` G ) ` x ) ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. (comp ` D ) ( ( 1st ` G ) ` x ) ) ( R ` x ) ) = ( R ` x ) )
33	14, 32	eqtrd 2508	. . 3 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( ( .1. o. ( 1st ` G ) ) ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. (comp ` D ) ( ( 1st ` G ) ` x ) ) ( R ` x ) ) = ( R ` x ) )
34	33	mpteq2dva 4539	. 2 |- ( ph -> ( x e. ( Base ` C ) |-> ( ( ( .1. o. ( 1st ` G ) ) ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. (comp ` D ) ( ( 1st ` G ) ` x ) ) ( R ` x ) ) ) = ( x e. ( Base ` C ) |-> ( R ` x ) ) )
35		fuclid.q	. . 3 |- Q = ( C FuncCat D )
36		fuclid.x	. . 3 |- .xb = (comp ` Q )
37	35, 5, 16, 8	fucidcl 15209	. . 3 |- ( ph -> ( .1. o. ( 1st ` G ) ) e. ( G N G ) )
38	35, 5, 1, 26, 36, 4, 37	fucco 15206	. 2 |- ( ph -> ( ( .1. o. ( 1st ` G ) ) ( <. F , G >. .xb G ) R ) = ( x e. ( Base ` C ) |-> ( ( ( .1. o. ( 1st ` G ) ) ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. (comp ` D ) ( ( 1st ` G ) ` x ) ) ( R ` x ) ) ) )
39	5, 28, 1	natfn 15198	. . 3 |- ( ph -> R Fn ( Base ` C ) )
40		dffn5 5919	. . 3 |- ( R Fn ( Base ` C ) <-> R = ( x e. ( Base ` C ) |-> ( R ` x ) ) )
41	39, 40	sylib 196	. 2 |- ( ph -> R = ( x e. ( Base ` C ) |-> ( R ` x ) ) )
42	34, 38, 41	3eqtr4d 2518	1 |- ( ph -> ( ( .1. o. ( 1st ` G ) ) ( <. F , G >. .xb G ) R ) = R )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1379   e. wcel 1767   <.cop 4039   class class class wbr 4453   |-> cmpt 4511   o. ccom 5009   Rel wrel 5010   Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295   1stc1st 6793   2ndc2nd 6794   Basecbs 14507   Hom chom 14583  compcco 14584   Catccat 14936   Idccid 14937   Func cfunc 15098   Nat cnat 15185   FuncCat cfuc 15186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-hom 14596  df-cco 14597  df-cat 14940  df-cid 14941  df-func 15102  df-nat 15187  df-fuc 15188

This theorem is referenced by:  fuccatid  15213