Theorem funcres2c 15119

Description: Condition for a functor to also be a functor into the restriction. (Contributed by Mario Carneiro, 30-Jan-2017.)

Hypotheses

Ref	Expression
funcres2c.a	|- A = ( Base ` C )
funcres2c.e	|- E = ( D ↾s S )
funcres2c.d	|- ( ph -> D e. Cat )
funcres2c.r	|- ( ph -> S e. V )
funcres2c.1	|- ( ph -> F : A --> S )

Assertion

Ref	Expression
funcres2c	|- ( ph -> ( F ( C Func D ) G <-> F ( C Func E ) G ) )

Proof of Theorem funcres2c

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

Step	Hyp	Ref	Expression
1		orc 385	. . 3 |- ( F ( C Func D ) G -> ( F ( C Func D ) G \/ F ( C Func E ) G ) )
2	1	a1i 11	. 2 |- ( ph -> ( F ( C Func D ) G -> ( F ( C Func D ) G \/ F ( C Func E ) G ) ) )
3		olc 384	. . 3 |- ( F ( C Func E ) G -> ( F ( C Func D ) G \/ F ( C Func E ) G ) )
4	3	a1i 11	. 2 |- ( ph -> ( F ( C Func E ) G -> ( F ( C Func D ) G \/ F ( C Func E ) G ) ) )
5		funcres2c.a	. . . . 5 |- A = ( Base ` C )
6		eqid 2462	. . . . 5 |- ( Hom ` C ) = ( Hom ` C )
7		eqid 2462	. . . . . . 7 |- ( Base ` D ) = ( Base ` D )
8		eqid 2462	. . . . . . 7 |- ( Hom f ` D ) = ( Hom f ` D )
9		funcres2c.d	. . . . . . 7 |- ( ph -> D e. Cat )
10		inss2 3714	. . . . . . . 8 |- ( S i^i ( Base ` D ) ) C_ ( Base ` D )
11	10	a1i 11	. . . . . . 7 |- ( ph -> ( S i^i ( Base ` D ) ) C_ ( Base ` D ) )
12	7, 8, 9, 11	fullsubc 15068	. . . . . 6 |- ( ph -> ( ( Hom f ` D ) |` ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) ) e. (Subcat ` D ) )
13	12	adantr 465	. . . . 5 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> ( ( Hom f ` D ) |` ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) ) e. (Subcat ` D ) )
14	8, 7	homffn 14940	. . . . . . 7 |- ( Hom f ` D ) Fn ( ( Base ` D ) X. ( Base ` D ) )
15		xpss12 5101	. . . . . . . 8 |- ( ( ( S i^i ( Base ` D ) ) C_ ( Base ` D ) /\ ( S i^i ( Base ` D ) ) C_ ( Base ` D ) ) -> ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) C_ ( ( Base ` D ) X. ( Base ` D ) ) )
16	10, 10, 15	mp2an 672	. . . . . . 7 |- ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) C_ ( ( Base ` D ) X. ( Base ` D ) )
17		fnssres 5687	. . . . . . 7 |- ( ( ( Hom f ` D ) Fn ( ( Base ` D ) X. ( Base ` D ) ) /\ ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) C_ ( ( Base ` D ) X. ( Base ` D ) ) ) -> ( ( Hom f ` D ) |` ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) ) Fn ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) )
18	14, 16, 17	mp2an 672	. . . . . 6 |- ( ( Hom f ` D ) |` ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) ) Fn ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) )
19	18	a1i 11	. . . . 5 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> ( ( Hom f ` D ) |` ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) ) Fn ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) )
20		funcres2c.1	. . . . . . . 8 |- ( ph -> F : A --> S )
21	20	adantr 465	. . . . . . 7 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> F : A --> S )
22		ffn 5724	. . . . . . 7 |- ( F : A --> S -> F Fn A )
23	21, 22	syl 16	. . . . . 6 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> F Fn A )
24		frn 5730	. . . . . . . 8 |- ( F : A --> S -> ran F C_ S )
25	21, 24	syl 16	. . . . . . 7 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> ran F C_ S )
26		simpr 461	. . . . . . . . . 10 |- ( ( ph /\ F ( C Func D ) G ) -> F ( C Func D ) G )
27	5, 7, 26	funcf1 15084	. . . . . . . . 9 |- ( ( ph /\ F ( C Func D ) G ) -> F : A --> ( Base ` D ) )
28		frn 5730	. . . . . . . . 9 |- ( F : A --> ( Base ` D ) -> ran F C_ ( Base ` D ) )
29	27, 28	syl 16	. . . . . . . 8 |- ( ( ph /\ F ( C Func D ) G ) -> ran F C_ ( Base ` D ) )
30		eqid 2462	. . . . . . . . . . 11 |- ( Base ` E ) = ( Base ` E )
31		simpr 461	. . . . . . . . . . 11 |- ( ( ph /\ F ( C Func E ) G ) -> F ( C Func E ) G )
32	5, 30, 31	funcf1 15084	. . . . . . . . . 10 |- ( ( ph /\ F ( C Func E ) G ) -> F : A --> ( Base ` E ) )
33		frn 5730	. . . . . . . . . 10 |- ( F : A --> ( Base ` E ) -> ran F C_ ( Base ` E ) )
34	32, 33	syl 16	. . . . . . . . 9 |- ( ( ph /\ F ( C Func E ) G ) -> ran F C_ ( Base ` E ) )
35		funcres2c.e	. . . . . . . . . 10 |- E = ( D ↾s S )
36	35, 7	ressbasss 14538	. . . . . . . . 9 |- ( Base ` E ) C_ ( Base ` D )
37	34, 36	syl6ss 3511	. . . . . . . 8 |- ( ( ph /\ F ( C Func E ) G ) -> ran F C_ ( Base ` D ) )
38	29, 37	jaodan 783	. . . . . . 7 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> ran F C_ ( Base ` D ) )
39	25, 38	ssind 3717	. . . . . 6 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> ran F C_ ( S i^i ( Base ` D ) ) )
40		df-f 5585	. . . . . 6 |- ( F : A --> ( S i^i ( Base ` D ) ) <-> ( F Fn A /\ ran F C_ ( S i^i ( Base ` D ) ) ) )
41	23, 39, 40	sylanbrc 664	. . . . 5 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> F : A --> ( S i^i ( Base ` D ) ) )
42		eqid 2462	. . . . . . . . 9 |- ( Hom ` D ) = ( Hom ` D )
43		simpr 461	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. A /\ y e. A ) ) /\ F ( C Func D ) G ) -> F ( C Func D ) G )
44		simplrl 759	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. A /\ y e. A ) ) /\ F ( C Func D ) G ) -> x e. A )
45		simplrr 760	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. A /\ y e. A ) ) /\ F ( C Func D ) G ) -> y e. A )
46	5, 6, 42, 43, 44, 45	funcf2 15086	. . . . . . . 8 |- ( ( ( ph /\ ( x e. A /\ y e. A ) ) /\ F ( C Func D ) G ) -> ( x G y ) : ( x ( Hom ` C ) y ) --> ( ( F ` x ) ( Hom ` D ) ( F ` y ) ) )
47		eqid 2462	. . . . . . . . . 10 |- ( Hom ` E ) = ( Hom ` E )
48		simpr 461	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. A /\ y e. A ) ) /\ F ( C Func E ) G ) -> F ( C Func E ) G )
49		simplrl 759	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. A /\ y e. A ) ) /\ F ( C Func E ) G ) -> x e. A )
50		simplrr 760	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. A /\ y e. A ) ) /\ F ( C Func E ) G ) -> y e. A )
51	5, 6, 47, 48, 49, 50	funcf2 15086	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. A /\ y e. A ) ) /\ F ( C Func E ) G ) -> ( x G y ) : ( x ( Hom ` C ) y ) --> ( ( F ` x ) ( Hom ` E ) ( F ` y ) ) )
52		eqidd 2463	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. A /\ y e. A ) ) /\ F ( C Func E ) G ) -> ( x ( Hom ` C ) y ) = ( x ( Hom ` C ) y ) )
53		funcres2c.r	. . . . . . . . . . . . 13 |- ( ph -> S e. V )
54	35, 42	resshom 14665	. . . . . . . . . . . . 13 |- ( S e. V -> ( Hom ` D ) = ( Hom ` E ) )
55	53, 54	syl 16	. . . . . . . . . . . 12 |- ( ph -> ( Hom ` D ) = ( Hom ` E ) )
56	55	ad2antrr 725	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. A /\ y e. A ) ) /\ F ( C Func E ) G ) -> ( Hom ` D ) = ( Hom ` E ) )
57	56	oveqd 6294	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. A /\ y e. A ) ) /\ F ( C Func E ) G ) -> ( ( F ` x ) ( Hom ` D ) ( F ` y ) ) = ( ( F ` x ) ( Hom ` E ) ( F ` y ) ) )
58	52, 57	feq23d 5719	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. A /\ y e. A ) ) /\ F ( C Func E ) G ) -> ( ( x G y ) : ( x ( Hom ` C ) y ) --> ( ( F ` x ) ( Hom ` D ) ( F ` y ) ) <-> ( x G y ) : ( x ( Hom ` C ) y ) --> ( ( F ` x ) ( Hom ` E ) ( F ` y ) ) ) )
59	51, 58	mpbird 232	. . . . . . . 8 |- ( ( ( ph /\ ( x e. A /\ y e. A ) ) /\ F ( C Func E ) G ) -> ( x G y ) : ( x ( Hom ` C ) y ) --> ( ( F ` x ) ( Hom ` D ) ( F ` y ) ) )
60	46, 59	jaodan 783	. . . . . . 7 |- ( ( ( ph /\ ( x e. A /\ y e. A ) ) /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> ( x G y ) : ( x ( Hom ` C ) y ) --> ( ( F ` x ) ( Hom ` D ) ( F ` y ) ) )
61	60	an32s 802	. . . . . 6 |- ( ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) /\ ( x e. A /\ y e. A ) ) -> ( x G y ) : ( x ( Hom ` C ) y ) --> ( ( F ` x ) ( Hom ` D ) ( F ` y ) ) )
62		eqidd 2463	. . . . . . 7 |- ( ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) /\ ( x e. A /\ y e. A ) ) -> ( x ( Hom ` C ) y ) = ( x ( Hom ` C ) y ) )
63	41	adantr 465	. . . . . . . . . 10 |- ( ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) /\ ( x e. A /\ y e. A ) ) -> F : A --> ( S i^i ( Base ` D ) ) )
64		simprl 755	. . . . . . . . . 10 |- ( ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) /\ ( x e. A /\ y e. A ) ) -> x e. A )
65	63, 64	ffvelrnd 6015	. . . . . . . . 9 |- ( ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) /\ ( x e. A /\ y e. A ) ) -> ( F ` x ) e. ( S i^i ( Base ` D ) ) )
66		simprr 756	. . . . . . . . . 10 |- ( ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) /\ ( x e. A /\ y e. A ) ) -> y e. A )
67	63, 66	ffvelrnd 6015	. . . . . . . . 9 |- ( ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) /\ ( x e. A /\ y e. A ) ) -> ( F ` y ) e. ( S i^i ( Base ` D ) ) )
68	65, 67	ovresd 6420	. . . . . . . 8 |- ( ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) /\ ( x e. A /\ y e. A ) ) -> ( ( F ` x ) ( ( Hom f ` D ) |` ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) ) ( F ` y ) ) = ( ( F ` x ) ( Hom f ` D ) ( F ` y ) ) )
69	10, 65	sseldi 3497	. . . . . . . . 9 |- ( ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) /\ ( x e. A /\ y e. A ) ) -> ( F ` x ) e. ( Base ` D ) )
70	10, 67	sseldi 3497	. . . . . . . . 9 |- ( ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) /\ ( x e. A /\ y e. A ) ) -> ( F ` y ) e. ( Base ` D ) )
71	8, 7, 42, 69, 70	homfval 14939	. . . . . . . 8 |- ( ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) /\ ( x e. A /\ y e. A ) ) -> ( ( F ` x ) ( Hom f ` D ) ( F ` y ) ) = ( ( F ` x ) ( Hom ` D ) ( F ` y ) ) )
72	68, 71	eqtrd 2503	. . . . . . 7 |- ( ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) /\ ( x e. A /\ y e. A ) ) -> ( ( F ` x ) ( ( Hom f ` D ) |` ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) ) ( F ` y ) ) = ( ( F ` x ) ( Hom ` D ) ( F ` y ) ) )
73	62, 72	feq23d 5719	. . . . . 6 |- ( ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) /\ ( x e. A /\ y e. A ) ) -> ( ( x G y ) : ( x ( Hom ` C ) y ) --> ( ( F ` x ) ( ( Hom f ` D ) |` ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) ) ( F ` y ) ) <-> ( x G y ) : ( x ( Hom ` C ) y ) --> ( ( F ` x ) ( Hom ` D ) ( F ` y ) ) ) )
74	61, 73	mpbird 232	. . . . 5 |- ( ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) /\ ( x e. A /\ y e. A ) ) -> ( x G y ) : ( x ( Hom ` C ) y ) --> ( ( F ` x ) ( ( Hom f ` D ) |` ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) ) ( F ` y ) ) )
75	5, 6, 13, 19, 41, 74	funcres2b 15115	. . . 4 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> ( F ( C Func D ) G <-> F ( C Func ( D |`cat ( ( Hom f ` D ) |` ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) ) ) ) G ) )
76		eqidd 2463	. . . . . 6 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> ( Hom f ` C ) = ( Hom f ` C ) )
77		eqidd 2463	. . . . . 6 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> (compf ` C ) = (compf ` C ) )
78	7	ressinbas 14542	. . . . . . . . . . 11 |- ( S e. V -> ( D ↾s S ) = ( D ↾s ( S i^i ( Base ` D ) ) ) )
79	53, 78	syl 16	. . . . . . . . . 10 |- ( ph -> ( D ↾s S ) = ( D ↾s ( S i^i ( Base ` D ) ) ) )
80	35, 79	syl5eq 2515	. . . . . . . . 9 |- ( ph -> E = ( D ↾s ( S i^i ( Base ` D ) ) ) )
81	80	fveq2d 5863	. . . . . . . 8 |- ( ph -> ( Hom f ` E ) = ( Hom f ` ( D ↾s ( S i^i ( Base ` D ) ) ) ) )
82		eqid 2462	. . . . . . . . . 10 |- ( D ↾s ( S i^i ( Base ` D ) ) ) = ( D ↾s ( S i^i ( Base ` D ) ) )
83		eqid 2462	. . . . . . . . . 10 |- ( D |`cat ( ( Hom f ` D ) |` ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) ) ) = ( D |`cat ( ( Hom f ` D ) |` ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) ) )
84	7, 8, 9, 11, 82, 83	fullresc 15069	. . . . . . . . 9 |- ( ph -> ( ( Hom f ` ( D ↾s ( S i^i ( Base ` D ) ) ) ) = ( Hom f ` ( D |`cat ( ( Hom f ` D ) |` ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) ) ) ) /\ (compf ` ( D ↾s ( S i^i ( Base ` D ) ) ) ) = (compf ` ( D |`cat ( ( Hom f ` D ) |` ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) ) ) ) ) )
85	84	simpld 459	. . . . . . . 8 |- ( ph -> ( Hom f ` ( D ↾s ( S i^i ( Base ` D ) ) ) ) = ( Hom f ` ( D |`cat ( ( Hom f ` D ) |` ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) ) ) ) )
86	81, 85	eqtrd 2503	. . . . . . 7 |- ( ph -> ( Hom f ` E ) = ( Hom f ` ( D |`cat ( ( Hom f ` D ) |` ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) ) ) ) )
87	86	adantr 465	. . . . . 6 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> ( Hom f ` E ) = ( Hom f ` ( D |`cat ( ( Hom f ` D ) |` ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) ) ) ) )
88	80	fveq2d 5863	. . . . . . . 8 |- ( ph -> (compf ` E ) = (compf ` ( D ↾s ( S i^i ( Base ` D ) ) ) ) )
89	84	simprd 463	. . . . . . . 8 |- ( ph -> (compf ` ( D ↾s ( S i^i ( Base ` D ) ) ) ) = (compf ` ( D |`cat ( ( Hom f ` D ) |` ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) ) ) ) )
90	88, 89	eqtrd 2503	. . . . . . 7 |- ( ph -> (compf ` E ) = (compf ` ( D |`cat ( ( Hom f ` D ) |` ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) ) ) ) )
91	90	adantr 465	. . . . . 6 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> (compf ` E ) = (compf ` ( D |`cat ( ( Hom f ` D ) |` ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) ) ) ) )
92		df-br 4443	. . . . . . . . . . 11 |- ( F ( C Func D ) G <-> <. F , G >. e. ( C Func D ) )
93		funcrcl 15081	. . . . . . . . . . 11 |- ( <. F , G >. e. ( C Func D ) -> ( C e. Cat /\ D e. Cat ) )
94	92, 93	sylbi 195	. . . . . . . . . 10 |- ( F ( C Func D ) G -> ( C e. Cat /\ D e. Cat ) )
95	94	simpld 459	. . . . . . . . 9 |- ( F ( C Func D ) G -> C e. Cat )
96		df-br 4443	. . . . . . . . . . 11 |- ( F ( C Func E ) G <-> <. F , G >. e. ( C Func E ) )
97		funcrcl 15081	. . . . . . . . . . 11 |- ( <. F , G >. e. ( C Func E ) -> ( C e. Cat /\ E e. Cat ) )
98	96, 97	sylbi 195	. . . . . . . . . 10 |- ( F ( C Func E ) G -> ( C e. Cat /\ E e. Cat ) )
99	98	simpld 459	. . . . . . . . 9 |- ( F ( C Func E ) G -> C e. Cat )
100	95, 99	jaoi 379	. . . . . . . 8 |- ( ( F ( C Func D ) G \/ F ( C Func E ) G ) -> C e. Cat )
101		elex 3117	. . . . . . . 8 |- ( C e. Cat -> C e. _V )
102	100, 101	syl 16	. . . . . . 7 |- ( ( F ( C Func D ) G \/ F ( C Func E ) G ) -> C e. _V )
103	102	adantl 466	. . . . . 6 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> C e. _V )
104		ovex 6302	. . . . . . . 8 |- ( D ↾s S ) e. _V
105	35, 104	eqeltri 2546	. . . . . . 7 |- E e. _V
106	105	a1i 11	. . . . . 6 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> E e. _V )
107		ovex 6302	. . . . . . 7 |- ( D |`cat ( ( Hom f ` D ) |` ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) ) ) e. _V
108	107	a1i 11	. . . . . 6 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> ( D |`cat ( ( Hom f ` D ) |` ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) ) ) e. _V )
109	76, 77, 87, 91, 103, 103, 106, 108	funcpropd 15118	. . . . 5 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> ( C Func E ) = ( C Func ( D |`cat ( ( Hom f ` D ) |` ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) ) ) ) )
110	109	breqd 4453	. . . 4 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> ( F ( C Func E ) G <-> F ( C Func ( D |`cat ( ( Hom f ` D ) |` ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) ) ) ) G ) )
111	75, 110	bitr4d 256	. . 3 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> ( F ( C Func D ) G <-> F ( C Func E ) G ) )
112	111	ex 434	. 2 |- ( ph -> ( ( F ( C Func D ) G \/ F ( C Func E ) G ) -> ( F ( C Func D ) G <-> F ( C Func E ) G ) ) )
113	2, 4, 112	pm5.21ndd 354	1 |- ( ph -> ( F ( C Func D ) G <-> F ( C Func E ) G ) )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1374   e. wcel 1762   _Vcvv 3108   i^i cin 3470   C_ wss 3471   <.cop 4028   class class class wbr 4442   X. cxp 4992   ran crn 4995   |` cres 4996   Fn wfn 5576   -->wf 5577   ` cfv 5581  (class class class)co 6277   Basecbs 14481   ↾_s cress 14482   Hom chom 14557   Catccat 14910   Hom _f chomf 14912  comp_fccomf 14913   |`_cat cresc 15029  Subcatcsubc 15030   Func cfunc 15072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7462  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-10 10593  df-n0 10787  df-z 10856  df-dec 10968  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-ress 14488  df-hom 14570  df-cco 14571  df-cat 14914  df-cid 14915  df-homf 14916  df-comf 14917  df-ssc 15031  df-resc 15032  df-subc 15033  df-func 15076

This theorem is referenced by:  fthres2c  15149  fullres2c  15157