Theorem cofth 16595

Description: The composition of two faithful functors is faithful. Proposition 3.30(c) in [Adamek] p. 35. (Contributed by Mario Carneiro, 28-Jan-2017.)

Hypotheses

Ref	Expression
cofth.f	|- ( ph -> F e. ( C Faith D ) )
cofth.g	|- ( ph -> G e. ( D Faith E ) )

Assertion

Ref	Expression
cofth	|- ( ph -> ( G o.func F ) e. ( C Faith E ) )

Proof of Theorem cofth

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

Step	Hyp	Ref	Expression
1		relfunc 16522	. . 3 |- Rel ( C Func E )
2		fthfunc 16567	. . . . 5 |- ( C Faith D ) C_ ( C Func D )
3		cofth.f	. . . . 5 |- ( ph -> F e. ( C Faith D ) )
4	2, 3	sseldi 3601	. . . 4 |- ( ph -> F e. ( C Func D ) )
5		fthfunc 16567	. . . . 5 |- ( D Faith E ) C_ ( D Func E )
6		cofth.g	. . . . 5 |- ( ph -> G e. ( D Faith E ) )
7	5, 6	sseldi 3601	. . . 4 |- ( ph -> G e. ( D Func E ) )
8	4, 7	cofucl 16548	. . 3 |- ( ph -> ( G o.func F ) e. ( C Func E ) )
9		1st2nd 7214	. . 3 |- ( ( Rel ( C Func E ) /\ ( G o.func F ) e. ( C Func E ) ) -> ( G o.func F ) = <. ( 1st ` ( G o.func F ) ) , ( 2nd ` ( G o.func F ) ) >. )
10	1, 8, 9	sylancr 695	. 2 |- ( ph -> ( G o.func F ) = <. ( 1st ` ( G o.func F ) ) , ( 2nd ` ( G o.func F ) ) >. )
11		1st2ndbr 7217	. . . . 5 |- ( ( Rel ( C Func E ) /\ ( G o.func F ) e. ( C Func E ) ) -> ( 1st ` ( G o.func F ) ) ( C Func E ) ( 2nd ` ( G o.func F ) ) )
12	1, 8, 11	sylancr 695	. . . 4 |- ( ph -> ( 1st ` ( G o.func F ) ) ( C Func E ) ( 2nd ` ( G o.func F ) ) )
13		eqid 2622	. . . . . . . 8 |- ( Base ` D ) = ( Base ` D )
14		eqid 2622	. . . . . . . 8 |- ( Hom ` D ) = ( Hom ` D )
15		eqid 2622	. . . . . . . 8 |- ( Hom ` E ) = ( Hom ` E )
16		relfth 16569	. . . . . . . . 9 |- Rel ( D Faith E )
17	6	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> G e. ( D Faith E ) )
18		1st2ndbr 7217	. . . . . . . . 9 |- ( ( Rel ( D Faith E ) /\ G e. ( D Faith E ) ) -> ( 1st ` G ) ( D Faith E ) ( 2nd ` G ) )
19	16, 17, 18	sylancr 695	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( 1st ` G ) ( D Faith E ) ( 2nd ` G ) )
20		eqid 2622	. . . . . . . . . 10 |- ( Base ` C ) = ( Base ` C )
21		relfunc 16522	. . . . . . . . . . 11 |- Rel ( C Func D )
22	4	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> F e. ( C Func D ) )
23		1st2ndbr 7217	. . . . . . . . . . 11 |- ( ( Rel ( C Func D ) /\ F e. ( C Func D ) ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
24	21, 22, 23	sylancr 695	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
25	20, 13, 24	funcf1 16526	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( 1st ` F ) : ( Base ` C ) --> ( Base ` D ) )
26		simprl 794	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> x e. ( Base ` C ) )
27	25, 26	ffvelrnd 6360	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( 1st ` F ) ` x ) e. ( Base ` D ) )
28		simprr 796	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> y e. ( Base ` C ) )
29	25, 28	ffvelrnd 6360	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( 1st ` F ) ` y ) e. ( Base ` D ) )
30	13, 14, 15, 19, 27, 29	fthf1 16577	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( ( 1st ` F ) ` x ) ( 2nd ` G ) ( ( 1st ` F ) ` y ) ) : ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) -1-1-> ( ( ( 1st ` G ) ` ( ( 1st ` F ) ` x ) ) ( Hom ` E ) ( ( 1st ` G ) ` ( ( 1st ` F ) ` y ) ) ) )
31		eqid 2622	. . . . . . . 8 |- ( Hom ` C ) = ( Hom ` C )
32		relfth 16569	. . . . . . . . 9 |- Rel ( C Faith D )
33	3	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> F e. ( C Faith D ) )
34		1st2ndbr 7217	. . . . . . . . 9 |- ( ( Rel ( C Faith D ) /\ F e. ( C Faith D ) ) -> ( 1st ` F ) ( C Faith D ) ( 2nd ` F ) )
35	32, 33, 34	sylancr 695	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( 1st ` F ) ( C Faith D ) ( 2nd ` F ) )
36	20, 31, 14, 35, 26, 28	fthf1 16577	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( x ( 2nd ` F ) y ) : ( x ( Hom ` C ) y ) -1-1-> ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) )
37		f1co 6110	. . . . . . 7 |- ( ( ( ( ( 1st ` F ) ` x ) ( 2nd ` G ) ( ( 1st ` F ) ` y ) ) : ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) -1-1-> ( ( ( 1st ` G ) ` ( ( 1st ` F ) ` x ) ) ( Hom ` E ) ( ( 1st ` G ) ` ( ( 1st ` F ) ` y ) ) ) /\ ( x ( 2nd ` F ) y ) : ( x ( Hom ` C ) y ) -1-1-> ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) ) -> ( ( ( ( 1st ` F ) ` x ) ( 2nd ` G ) ( ( 1st ` F ) ` y ) ) o. ( x ( 2nd ` F ) y ) ) : ( x ( Hom ` C ) y ) -1-1-> ( ( ( 1st ` G ) ` ( ( 1st ` F ) ` x ) ) ( Hom ` E ) ( ( 1st ` G ) ` ( ( 1st ` F ) ` y ) ) ) )
38	30, 36, 37	syl2anc 693	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( ( ( 1st ` F ) ` x ) ( 2nd ` G ) ( ( 1st ` F ) ` y ) ) o. ( x ( 2nd ` F ) y ) ) : ( x ( Hom ` C ) y ) -1-1-> ( ( ( 1st ` G ) ` ( ( 1st ` F ) ` x ) ) ( Hom ` E ) ( ( 1st ` G ) ` ( ( 1st ` F ) ` y ) ) ) )
39	7	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> G e. ( D Func E ) )
40	20, 22, 39, 26, 28	cofu2nd 16545	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( x ( 2nd ` ( G o.func F ) ) y ) = ( ( ( ( 1st ` F ) ` x ) ( 2nd ` G ) ( ( 1st ` F ) ` y ) ) o. ( x ( 2nd ` F ) y ) ) )
41		eqidd 2623	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( x ( Hom ` C ) y ) = ( x ( Hom ` C ) y ) )
42	20, 22, 39, 26	cofu1 16544	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( 1st ` ( G o.func F ) ) ` x ) = ( ( 1st ` G ) ` ( ( 1st ` F ) ` x ) ) )
43	20, 22, 39, 28	cofu1 16544	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( 1st ` ( G o.func F ) ) ` y ) = ( ( 1st ` G ) ` ( ( 1st ` F ) ` y ) ) )
44	42, 43	oveq12d 6668	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( ( 1st ` ( G o.func F ) ) ` x ) ( Hom ` E ) ( ( 1st ` ( G o.func F ) ) ` y ) ) = ( ( ( 1st ` G ) ` ( ( 1st ` F ) ` x ) ) ( Hom ` E ) ( ( 1st ` G ) ` ( ( 1st ` F ) ` y ) ) ) )
45	40, 41, 44	f1eq123d 6131	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( ( x ( 2nd ` ( G o.func F ) ) y ) : ( x ( Hom ` C ) y ) -1-1-> ( ( ( 1st ` ( G o.func F ) ) ` x ) ( Hom ` E ) ( ( 1st ` ( G o.func F ) ) ` y ) ) <-> ( ( ( ( 1st ` F ) ` x ) ( 2nd ` G ) ( ( 1st ` F ) ` y ) ) o. ( x ( 2nd ` F ) y ) ) : ( x ( Hom ` C ) y ) -1-1-> ( ( ( 1st ` G ) ` ( ( 1st ` F ) ` x ) ) ( Hom ` E ) ( ( 1st ` G ) ` ( ( 1st ` F ) ` y ) ) ) ) )
46	38, 45	mpbird 247	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) ) ) -> ( x ( 2nd ` ( G o.func F ) ) y ) : ( x ( Hom ` C ) y ) -1-1-> ( ( ( 1st ` ( G o.func F ) ) ` x ) ( Hom ` E ) ( ( 1st ` ( G o.func F ) ) ` y ) ) )
47	46	ralrimivva 2971	. . . 4 |- ( ph -> A. x e. ( Base ` C ) A. y e. ( Base ` C ) ( x ( 2nd ` ( G o.func F ) ) y ) : ( x ( Hom ` C ) y ) -1-1-> ( ( ( 1st ` ( G o.func F ) ) ` x ) ( Hom ` E ) ( ( 1st ` ( G o.func F ) ) ` y ) ) )
48	20, 31, 15	isfth2 16575	. . . 4 |- ( ( 1st ` ( G o.func F ) ) ( C Faith E ) ( 2nd ` ( G o.func F ) ) <-> ( ( 1st ` ( G o.func F ) ) ( C Func E ) ( 2nd ` ( G o.func F ) ) /\ A. x e. ( Base ` C ) A. y e. ( Base ` C ) ( x ( 2nd ` ( G o.func F ) ) y ) : ( x ( Hom ` C ) y ) -1-1-> ( ( ( 1st ` ( G o.func F ) ) ` x ) ( Hom ` E ) ( ( 1st ` ( G o.func F ) ) ` y ) ) ) )
49	12, 47, 48	sylanbrc 698	. . 3 |- ( ph -> ( 1st ` ( G o.func F ) ) ( C Faith E ) ( 2nd ` ( G o.func F ) ) )
50		df-br 4654	. . 3 |- ( ( 1st ` ( G o.func F ) ) ( C Faith E ) ( 2nd ` ( G o.func F ) ) <-> <. ( 1st ` ( G o.func F ) ) , ( 2nd ` ( G o.func F ) ) >. e. ( C Faith E ) )
51	49, 50	sylib 208	. 2 |- ( ph -> <. ( 1st ` ( G o.func F ) ) , ( 2nd ` ( G o.func F ) ) >. e. ( C Faith E ) )
52	10, 51	eqeltrd 2701	1 |- ( ph -> ( G o.func F ) e. ( C Faith E ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   <.cop 4183   class class class wbr 4653   o. ccom 5118   Rel wrel 5119   -1-1->wf1 5885   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   Basecbs 15857   Hom chom 15952   Func cfunc 16514   o._func ccofu 16516   Faith cfth 16563

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-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-rmo 2920  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-ixp 7909  df-cat 16329  df-cid 16330  df-func 16518  df-cofu 16520  df-fth 16565

This theorem is referenced by:  coffth  16596