Theorem fuccocl 16624

Description: The composition of two natural transformations is a natural transformation. Remark 6.14(a) in [Adamek] p. 87. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
fuccocl.q	|- Q = ( C FuncCat D )
fuccocl.n	|- N = ( C Nat D )
fuccocl.x	|- .xb = (comp ` Q )
fuccocl.r	|- ( ph -> R e. ( F N G ) )
fuccocl.s	|- ( ph -> S e. ( G N H ) )

Assertion

Ref	Expression
fuccocl	|- ( ph -> ( S ( <. F , G >. .xb H ) R ) e. ( F N H ) )

Proof of Theorem fuccocl

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

Step	Hyp	Ref	Expression
1		fuccocl.q	. . . 4 |- Q = ( C FuncCat D )
2		fuccocl.n	. . . 4 |- N = ( C Nat D )
3		eqid 2622	. . . 4 |- ( Base ` C ) = ( Base ` C )
4		eqid 2622	. . . 4 |- (comp ` D ) = (comp ` D )
5		fuccocl.x	. . . 4 |- .xb = (comp ` Q )
6		fuccocl.r	. . . 4 |- ( ph -> R e. ( F N G ) )
7		fuccocl.s	. . . 4 |- ( ph -> S e. ( G N H ) )
8	1, 2, 3, 4, 5, 6, 7	fucco 16622	. . 3 |- ( ph -> ( S ( <. F , G >. .xb H ) R ) = ( x e. ( Base ` C ) |-> ( ( S ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. (comp ` D ) ( ( 1st ` H ) ` x ) ) ( R ` x ) ) ) )
9		eqid 2622	. . . . . 6 |- ( Base ` D ) = ( Base ` D )
10		eqid 2622	. . . . . 6 |- ( Hom ` D ) = ( Hom ` D )
11	2	natrcl 16610	. . . . . . . . . . 11 |- ( R e. ( F N G ) -> ( F e. ( C Func D ) /\ G e. ( C Func D ) ) )
12	6, 11	syl 17	. . . . . . . . . 10 |- ( ph -> ( F e. ( C Func D ) /\ G e. ( C Func D ) ) )
13	12	simpld 475	. . . . . . . . 9 |- ( ph -> F e. ( C Func D ) )
14		funcrcl 16523	. . . . . . . . 9 |- ( F e. ( C Func D ) -> ( C e. Cat /\ D e. Cat ) )
15	13, 14	syl 17	. . . . . . . 8 |- ( ph -> ( C e. Cat /\ D e. Cat ) )
16	15	simprd 479	. . . . . . 7 |- ( ph -> D e. Cat )
17	16	adantr 481	. . . . . 6 |- ( ( ph /\ x e. ( Base ` C ) ) -> D e. Cat )
18		relfunc 16522	. . . . . . . . 9 |- Rel ( C Func D )
19		1st2ndbr 7217	. . . . . . . . 9 |- ( ( Rel ( C Func D ) /\ F e. ( C Func D ) ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
20	18, 13, 19	sylancr 695	. . . . . . . 8 |- ( ph -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
21	3, 9, 20	funcf1 16526	. . . . . . 7 |- ( ph -> ( 1st ` F ) : ( Base ` C ) --> ( Base ` D ) )
22	21	ffvelrnda 6359	. . . . . 6 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` F ) ` x ) e. ( Base ` D ) )
23	2	natrcl 16610	. . . . . . . . . . 11 |- ( S e. ( G N H ) -> ( G e. ( C Func D ) /\ H e. ( C Func D ) ) )
24	7, 23	syl 17	. . . . . . . . . 10 |- ( ph -> ( G e. ( C Func D ) /\ H e. ( C Func D ) ) )
25	24	simpld 475	. . . . . . . . 9 |- ( ph -> G e. ( C Func D ) )
26		1st2ndbr 7217	. . . . . . . . 9 |- ( ( Rel ( C Func D ) /\ G e. ( C Func D ) ) -> ( 1st ` G ) ( C Func D ) ( 2nd ` G ) )
27	18, 25, 26	sylancr 695	. . . . . . . 8 |- ( ph -> ( 1st ` G ) ( C Func D ) ( 2nd ` G ) )
28	3, 9, 27	funcf1 16526	. . . . . . 7 |- ( ph -> ( 1st ` G ) : ( Base ` C ) --> ( Base ` D ) )
29	28	ffvelrnda 6359	. . . . . 6 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` G ) ` x ) e. ( Base ` D ) )
30	24	simprd 479	. . . . . . . . 9 |- ( ph -> H e. ( C Func D ) )
31		1st2ndbr 7217	. . . . . . . . 9 |- ( ( Rel ( C Func D ) /\ H e. ( C Func D ) ) -> ( 1st ` H ) ( C Func D ) ( 2nd ` H ) )
32	18, 30, 31	sylancr 695	. . . . . . . 8 |- ( ph -> ( 1st ` H ) ( C Func D ) ( 2nd ` H ) )
33	3, 9, 32	funcf1 16526	. . . . . . 7 |- ( ph -> ( 1st ` H ) : ( Base ` C ) --> ( Base ` D ) )
34	33	ffvelrnda 6359	. . . . . 6 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( 1st ` H ) ` x ) e. ( Base ` D ) )
35	2, 6	nat1st2nd 16611	. . . . . . . 8 |- ( ph -> R e. ( <. ( 1st ` F ) , ( 2nd ` F ) >. N <. ( 1st ` G ) , ( 2nd ` G ) >. ) )
36	35	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` C ) ) -> R e. ( <. ( 1st ` F ) , ( 2nd ` F ) >. N <. ( 1st ` G ) , ( 2nd ` G ) >. ) )
37		simpr 477	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` C ) ) -> x e. ( Base ` C ) )
38	2, 36, 3, 10, 37	natcl 16613	. . . . . 6 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( R ` x ) e. ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` G ) ` x ) ) )
39	2, 7	nat1st2nd 16611	. . . . . . . 8 |- ( ph -> S e. ( <. ( 1st ` G ) , ( 2nd ` G ) >. N <. ( 1st ` H ) , ( 2nd ` H ) >. ) )
40	39	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. ( Base ` C ) ) -> S e. ( <. ( 1st ` G ) , ( 2nd ` G ) >. N <. ( 1st ` H ) , ( 2nd ` H ) >. ) )
41	2, 40, 3, 10, 37	natcl 16613	. . . . . 6 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( S ` x ) e. ( ( ( 1st ` G ) ` x ) ( Hom ` D ) ( ( 1st ` H ) ` x ) ) )
42	9, 10, 4, 17, 22, 29, 34, 38, 41	catcocl 16346	. . . . 5 |- ( ( ph /\ x e. ( Base ` C ) ) -> ( ( S ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. (comp ` D ) ( ( 1st ` H ) ` x ) ) ( R ` x ) ) e. ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` H ) ` x ) ) )
43	42	ralrimiva 2966	. . . 4 |- ( ph -> A. x e. ( Base ` C ) ( ( S ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. (comp ` D ) ( ( 1st ` H ) ` x ) ) ( R ` x ) ) e. ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` H ) ` x ) ) )
44		fvex 6201	. . . . 5 |- ( Base ` C ) e. _V
45		mptelixpg 7945	. . . . 5 |- ( ( Base ` C ) e. _V -> ( ( x e. ( Base ` C ) |-> ( ( S ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. (comp ` D ) ( ( 1st ` H ) ` x ) ) ( R ` x ) ) ) e. X_ x e. ( Base ` C ) ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` H ) ` x ) ) <-> A. x e. ( Base ` C ) ( ( S ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. (comp ` D ) ( ( 1st ` H ) ` x ) ) ( R ` x ) ) e. ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` H ) ` x ) ) ) )
46	44, 45	ax-mp 5	. . . 4 |- ( ( x e. ( Base ` C ) |-> ( ( S ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. (comp ` D ) ( ( 1st ` H ) ` x ) ) ( R ` x ) ) ) e. X_ x e. ( Base ` C ) ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` H ) ` x ) ) <-> A. x e. ( Base ` C ) ( ( S ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. (comp ` D ) ( ( 1st ` H ) ` x ) ) ( R ` x ) ) e. ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` H ) ` x ) ) )
47	43, 46	sylibr 224	. . 3 |- ( ph -> ( x e. ( Base ` C ) |-> ( ( S ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. (comp ` D ) ( ( 1st ` H ) ` x ) ) ( R ` x ) ) ) e. X_ x e. ( Base ` C ) ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` H ) ` x ) ) )
48	8, 47	eqeltrd 2701	. 2 |- ( ph -> ( S ( <. F , G >. .xb H ) R ) e. X_ x e. ( Base ` C ) ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` H ) ` x ) ) )
49	16	adantr 481	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> D e. Cat )
50	21	adantr 481	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( 1st ` F ) : ( Base ` C ) --> ( Base ` D ) )
51		simpr1 1067	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> x e. ( Base ` C ) )
52	50, 51	ffvelrnd 6360	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( 1st ` F ) ` x ) e. ( Base ` D ) )
53		simpr2 1068	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> y e. ( Base ` C ) )
54	50, 53	ffvelrnd 6360	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( 1st ` F ) ` y ) e. ( Base ` D ) )
55	28	adantr 481	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( 1st ` G ) : ( Base ` C ) --> ( Base ` D ) )
56	55, 53	ffvelrnd 6360	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( 1st ` G ) ` y ) e. ( Base ` D ) )
57		eqid 2622	. . . . . . . 8 |- ( Hom ` C ) = ( Hom ` C )
58	20	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( 1st ` F ) ( C Func D ) ( 2nd ` F ) )
59	3, 57, 10, 58, 51, 53	funcf2 16528	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( x ( 2nd ` F ) y ) : ( x ( Hom ` C ) y ) --> ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) )
60		simpr3 1069	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> f e. ( x ( Hom ` C ) y ) )
61	59, 60	ffvelrnd 6360	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( x ( 2nd ` F ) y ) ` f ) e. ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` F ) ` y ) ) )
62	35	adantr 481	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> R e. ( <. ( 1st ` F ) , ( 2nd ` F ) >. N <. ( 1st ` G ) , ( 2nd ` G ) >. ) )
63	2, 62, 3, 10, 53	natcl 16613	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( R ` y ) e. ( ( ( 1st ` F ) ` y ) ( Hom ` D ) ( ( 1st ` G ) ` y ) ) )
64	33	adantr 481	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( 1st ` H ) : ( Base ` C ) --> ( Base ` D ) )
65	64, 53	ffvelrnd 6360	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( 1st ` H ) ` y ) e. ( Base ` D ) )
66	39	adantr 481	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> S e. ( <. ( 1st ` G ) , ( 2nd ` G ) >. N <. ( 1st ` H ) , ( 2nd ` H ) >. ) )
67	2, 66, 3, 10, 53	natcl 16613	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( S ` y ) e. ( ( ( 1st ` G ) ` y ) ( Hom ` D ) ( ( 1st ` H ) ` y ) ) )
68	9, 10, 4, 49, 52, 54, 56, 61, 63, 65, 67	catass 16347	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( ( S ` y ) ( <. ( ( 1st ` F ) ` y ) , ( ( 1st ` G ) ` y ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( R ` y ) ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) = ( ( S ` y ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` y ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( R ` y ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. (comp ` D ) ( ( 1st ` G ) ` y ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) ) )
69	2, 62, 3, 57, 4, 51, 53, 60	nati 16615	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( R ` y ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. (comp ` D ) ( ( 1st ` G ) ` y ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) = ( ( ( x ( 2nd ` G ) y ) ` f ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. (comp ` D ) ( ( 1st ` G ) ` y ) ) ( R ` x ) ) )
70	69	oveq2d 6666	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( S ` y ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` y ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( R ` y ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. (comp ` D ) ( ( 1st ` G ) ` y ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) ) = ( ( S ` y ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` y ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( ( x ( 2nd ` G ) y ) ` f ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. (comp ` D ) ( ( 1st ` G ) ` y ) ) ( R ` x ) ) ) )
71	55, 51	ffvelrnd 6360	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( 1st ` G ) ` x ) e. ( Base ` D ) )
72	2, 62, 3, 10, 51	natcl 16613	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( R ` x ) e. ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` G ) ` x ) ) )
73	27	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( 1st ` G ) ( C Func D ) ( 2nd ` G ) )
74	3, 57, 10, 73, 51, 53	funcf2 16528	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( x ( 2nd ` G ) y ) : ( x ( Hom ` C ) y ) --> ( ( ( 1st ` G ) ` x ) ( Hom ` D ) ( ( 1st ` G ) ` y ) ) )
75	74, 60	ffvelrnd 6360	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( x ( 2nd ` G ) y ) ` f ) e. ( ( ( 1st ` G ) ` x ) ( Hom ` D ) ( ( 1st ` G ) ` y ) ) )
76	9, 10, 4, 49, 52, 71, 56, 72, 75, 65, 67	catass 16347	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( ( S ` y ) ( <. ( ( 1st ` G ) ` x ) , ( ( 1st ` G ) ` y ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( x ( 2nd ` G ) y ) ` f ) ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( R ` x ) ) = ( ( S ` y ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` y ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( ( x ( 2nd ` G ) y ) ` f ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. (comp ` D ) ( ( 1st ` G ) ` y ) ) ( R ` x ) ) ) )
77	2, 66, 3, 57, 4, 51, 53, 60	nati 16615	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( S ` y ) ( <. ( ( 1st ` G ) ` x ) , ( ( 1st ` G ) ` y ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( x ( 2nd ` G ) y ) ` f ) ) = ( ( ( x ( 2nd ` H ) y ) ` f ) ( <. ( ( 1st ` G ) ` x ) , ( ( 1st ` H ) ` x ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( S ` x ) ) )
78	77	oveq1d 6665	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( ( S ` y ) ( <. ( ( 1st ` G ) ` x ) , ( ( 1st ` G ) ` y ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( x ( 2nd ` G ) y ) ` f ) ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( R ` x ) ) = ( ( ( ( x ( 2nd ` H ) y ) ` f ) ( <. ( ( 1st ` G ) ` x ) , ( ( 1st ` H ) ` x ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( S ` x ) ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( R ` x ) ) )
79	70, 76, 78	3eqtr2d 2662	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( S ` y ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` y ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( R ` y ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. (comp ` D ) ( ( 1st ` G ) ` y ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) ) = ( ( ( ( x ( 2nd ` H ) y ) ` f ) ( <. ( ( 1st ` G ) ` x ) , ( ( 1st ` H ) ` x ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( S ` x ) ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( R ` x ) ) )
80	64, 51	ffvelrnd 6360	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( 1st ` H ) ` x ) e. ( Base ` D ) )
81	2, 66, 3, 10, 51	natcl 16613	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( S ` x ) e. ( ( ( 1st ` G ) ` x ) ( Hom ` D ) ( ( 1st ` H ) ` x ) ) )
82	32	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( 1st ` H ) ( C Func D ) ( 2nd ` H ) )
83	3, 57, 10, 82, 51, 53	funcf2 16528	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( x ( 2nd ` H ) y ) : ( x ( Hom ` C ) y ) --> ( ( ( 1st ` H ) ` x ) ( Hom ` D ) ( ( 1st ` H ) ` y ) ) )
84	83, 60	ffvelrnd 6360	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( x ( 2nd ` H ) y ) ` f ) e. ( ( ( 1st ` H ) ` x ) ( Hom ` D ) ( ( 1st ` H ) ` y ) ) )
85	9, 10, 4, 49, 52, 71, 80, 72, 81, 65, 84	catass 16347	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( ( ( x ( 2nd ` H ) y ) ` f ) ( <. ( ( 1st ` G ) ` x ) , ( ( 1st ` H ) ` x ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( S ` x ) ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( R ` x ) ) = ( ( ( x ( 2nd ` H ) y ) ` f ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` H ) ` x ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( S ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. (comp ` D ) ( ( 1st ` H ) ` x ) ) ( R ` x ) ) ) )
86	68, 79, 85	3eqtrd 2660	. . . 4 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( ( S ` y ) ( <. ( ( 1st ` F ) ` y ) , ( ( 1st ` G ) ` y ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( R ` y ) ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) = ( ( ( x ( 2nd ` H ) y ) ` f ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` H ) ` x ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( S ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. (comp ` D ) ( ( 1st ` H ) ` x ) ) ( R ` x ) ) ) )
87	6	adantr 481	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> R e. ( F N G ) )
88	7	adantr 481	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> S e. ( G N H ) )
89	1, 2, 3, 4, 5, 87, 88, 53	fuccoval 16623	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( S ( <. F , G >. .xb H ) R ) ` y ) = ( ( S ` y ) ( <. ( ( 1st ` F ) ` y ) , ( ( 1st ` G ) ` y ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( R ` y ) ) )
90	89	oveq1d 6665	. . . 4 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( ( S ( <. F , G >. .xb H ) R ) ` y ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) = ( ( ( S ` y ) ( <. ( ( 1st ` F ) ` y ) , ( ( 1st ` G ) ` y ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( R ` y ) ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) )
91	1, 2, 3, 4, 5, 87, 88, 51	fuccoval 16623	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( S ( <. F , G >. .xb H ) R ) ` x ) = ( ( S ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. (comp ` D ) ( ( 1st ` H ) ` x ) ) ( R ` x ) ) )
92	91	oveq2d 6666	. . . 4 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( ( x ( 2nd ` H ) y ) ` f ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` H ) ` x ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( S ( <. F , G >. .xb H ) R ) ` x ) ) = ( ( ( x ( 2nd ` H ) y ) ` f ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` H ) ` x ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( S ` x ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` G ) ` x ) >. (comp ` D ) ( ( 1st ` H ) ` x ) ) ( R ` x ) ) ) )
93	86, 90, 92	3eqtr4d 2666	. . 3 |- ( ( ph /\ ( x e. ( Base ` C ) /\ y e. ( Base ` C ) /\ f e. ( x ( Hom ` C ) y ) ) ) -> ( ( ( S ( <. F , G >. .xb H ) R ) ` y ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) = ( ( ( x ( 2nd ` H ) y ) ` f ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` H ) ` x ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( S ( <. F , G >. .xb H ) R ) ` x ) ) )
94	93	ralrimivvva 2972	. 2 |- ( ph -> A. x e. ( Base ` C ) A. y e. ( Base ` C ) A. f e. ( x ( Hom ` C ) y ) ( ( ( S ( <. F , G >. .xb H ) R ) ` y ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) = ( ( ( x ( 2nd ` H ) y ) ` f ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` H ) ` x ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( S ( <. F , G >. .xb H ) R ) ` x ) ) )
95	2, 3, 57, 10, 4, 13, 30	isnat2 16608	. 2 |- ( ph -> ( ( S ( <. F , G >. .xb H ) R ) e. ( F N H ) <-> ( ( S ( <. F , G >. .xb H ) R ) e. X_ x e. ( Base ` C ) ( ( ( 1st ` F ) ` x ) ( Hom ` D ) ( ( 1st ` H ) ` x ) ) /\ A. x e. ( Base ` C ) A. y e. ( Base ` C ) A. f e. ( x ( Hom ` C ) y ) ( ( ( S ( <. F , G >. .xb H ) R ) ` y ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` F ) ` y ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( x ( 2nd ` F ) y ) ` f ) ) = ( ( ( x ( 2nd ` H ) y ) ` f ) ( <. ( ( 1st ` F ) ` x ) , ( ( 1st ` H ) ` x ) >. (comp ` D ) ( ( 1st ` H ) ` y ) ) ( ( S ( <. F , G >. .xb H ) R ) ` x ) ) ) ) )
96	48, 94, 95	mpbir2and 957	1 |- ( ph -> ( S ( <. F , G >. .xb H ) R ) e. ( F N H ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   Rel wrel 5119   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   X_cixp 7908   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-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-map 7859  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-cat 16329  df-func 16518  df-nat 16603  df-fuc 16604

This theorem is referenced by:  fucass  16628  fuccatid  16629  evlfcllem  16861  yonedalem3b  16919