Theorem fuclid 15883

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 2453	. . . . . . 7 |- ( Base ` C ) = ( Base ` C )
2		eqid 2453	. . . . . . 7 |- ( Base ` D ) = ( Base ` D )
3		relfunc 15779	. . . . . . . 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 15867	. . . . . . . . . 10 |- ( R e. ( F N G ) -> ( F e. ( C Func D ) /\ G e. ( C Func D ) ) )
7	4, 6	syl 17	. . . . . . . . 9 |- ( ph -> ( F e. ( C Func D ) /\ G e. ( C Func D ) ) )
8	7	simprd 465	. . . . . . . 8 |- ( ph -> G e. ( C Func D ) )
9		1st2ndbr 6847	. . . . . . . 8 |- ( ( Rel ( C Func D ) /\ G e. ( C Func D ) ) -> ( 1st ` G ) ( C Func D ) ( 2nd ` G ) )
10	3, 8, 9	sylancr 670	. . . . . . 7 |- ( ph -> ( 1st ` G ) ( C Func D ) ( 2nd ` G ) )
11	1, 2, 10	funcf1 15783	. . . . . 6 |- ( ph -> ( 1st ` G ) : ( Base ` C ) --> ( Base ` D ) )
12		fvco3 5947	. . . . . 6 |- ( ( ( 1st ` G ) : ( Base ` C ) --> ( Base ` D ) /\ x e. ( Base ` C ) ) -> ( ( .1. o. ( 1st ` G ) ) ` x ) = ( .1. ` ( ( 1st ` G ) ` x ) ) )
13	11, 12	sylan 474	. . . . 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 2453	. . . . 5 |- ( Hom ` D ) = ( Hom ` D )
16		fuclid.1	. . . . 5 |- .1. = ( Id ` D )
17	7	simpld 461	. . . . . . . 8 |- ( ph -> F e. ( C Func D ) )
18		funcrcl 15780	. . . . . . . 8 |- ( F e. ( C Func D ) -> ( C e. Cat /\ D e. Cat ) )
19	17, 18	syl 17	. . . . . . 7 |- ( ph -> ( C e. Cat /\ D e. Cat ) )
20	19	simprd 465	. . . . . 6 |- ( ph -> D e. Cat )
21	20	adantr 467	. . . . 5 |- ( ( ph /\ x e. ( Base ` C ) ) -> D e. Cat )
22		1st2ndbr 6847	. . . . . . . 8 |- ( ( Rel ( C Func D ) /\ F e. ( C Func D ) ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
23	3, 17, 22	sylancr 670	. . . . . . 7 |- ( ph -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
24	1, 2, 23	funcf1 15783	. . . . . 6 |- ( ph -> ( 1st ` F ) : ( Base ` C ) --> ( Base ` D ) )
25	24	ffvelrnda 6027	. . . . 5 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` F ) ` x ) e. ( Base ` D ) )
26		eqid 2453	. . . . 5 |- (comp ` D ) = (comp ` D )
27	11	ffvelrnda 6027	. . . . 5 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` G ) ` x ) e. ( Base ` D ) )
28	5, 4	nat1st2nd 15868	. . . . . . 7 |- ( ph -> R e. ( <. ( 1st ` F ) , ( 2nd ` F ) >. N <. ( 1st ` G ) , ( 2nd ` G ) >. ) )
29	28	adantr 467	. . . . . 6 |- ( ( ph /\ x e. ( Base ` C ) ) -> R e. ( <. ( 1st ` F ) , ( 2nd ` F ) >. N <. ( 1st ` G ) , ( 2nd ` G ) >. ) )
30		simpr 463	. . . . . 6 |- ( ( ph /\ x e. ( Base ` C ) ) -> x e. ( Base ` C ) )
31	5, 29, 1, 15, 30	natcl 15870	. . . . 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 15601	. . . 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 2487	. . 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 4492	. 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 15882	. . 3 |- ( ph -> ( .1. o. ( 1st ` G ) ) e. ( G N G ) )
38	35, 5, 1, 26, 36, 4, 37	fucco 15879	. 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 15871	. . 3 |- ( ph -> R Fn ( Base ` C ) )
40		dffn5 5915	. . 3 |- ( R Fn ( Base ` C ) <-> R = ( x e. ( Base ` C ) |-> ( R ` x ) ) )
41	39, 40	sylib 200	. 2 |- ( ph -> R = ( x e. ( Base ` C ) |-> ( R ` x ) ) )
42	34, 38, 41	3eqtr4d 2497	1 |- ( ph -> ( ( .1. o. ( 1st ` G ) ) ( <. F , G >. .xb G ) R ) = R )

Syntax hints:   -> wi 4   /\ wa 371   = wceq 1446   e. wcel 1889   <.cop 3976   class class class wbr 4405   |-> cmpt 4464   o. ccom 4841   Rel wrel 4842   Fn wfn 5580   -->wf 5581   ` cfv 5585  (class class class)co 6295   1stc1st 6796   2ndc2nd 6797   Basecbs 15133   Hom chom 15213  compcco 15214   Catccat 15582   Idccid 15583   Func cfunc 15771   Nat cnat 15858   FuncCat cfuc 15859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-oadd 7191  df-er 7368  df-map 7479  df-ixp 7528  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-7 10680  df-8 10681  df-9 10682  df-10 10683  df-n0 10877  df-z 10945  df-dec 11059  df-uz 11167  df-fz 11792  df-struct 15135  df-ndx 15136  df-slot 15137  df-base 15138  df-hom 15226  df-cco 15227  df-cat 15586  df-cid 15587  df-func 15775  df-nat 15860  df-fuc 15861

This theorem is referenced by:  fuccatid  15886