Theorem funcres2c 15392

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 383	. . 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 382	. . 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 2454	. . . . 5 |- ( Hom ` C ) = ( Hom ` C )
7		eqid 2454	. . . . . . 7 |- ( Base ` D ) = ( Base ` D )
8		eqid 2454	. . . . . . 7 |- ( Hom f ` D ) = ( Hom f ` D )
9		funcres2c.d	. . . . . . 7 |- ( ph -> D e. Cat )
10		inss2 3705	. . . . . . . 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 15341	. . . . . 6 |- ( ph -> ( ( Hom f ` D ) |` ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) ) e. (Subcat ` D ) )
13	12	adantr 463	. . . . 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 15184	. . . . . . 7 |- ( Hom f ` D ) Fn ( ( Base ` D ) X. ( Base ` D ) )
15		xpss12 5096	. . . . . . . 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 670	. . . . . . 7 |- ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) C_ ( ( Base ` D ) X. ( Base ` D ) )
17		fnssres 5676	. . . . . . 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 670	. . . . . 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 463	. . . . . . 7 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> F : A --> S )
22		ffn 5713	. . . . . . 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 5719	. . . . . . . 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 459	. . . . . . . . . 10 |- ( ( ph /\ F ( C Func D ) G ) -> F ( C Func D ) G )
27	5, 7, 26	funcf1 15357	. . . . . . . . 9 |- ( ( ph /\ F ( C Func D ) G ) -> F : A --> ( Base ` D ) )
28		frn 5719	. . . . . . . . 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 2454	. . . . . . . . . . 11 |- ( Base ` E ) = ( Base ` E )
31		simpr 459	. . . . . . . . . . 11 |- ( ( ph /\ F ( C Func E ) G ) -> F ( C Func E ) G )
32	5, 30, 31	funcf1 15357	. . . . . . . . . 10 |- ( ( ph /\ F ( C Func E ) G ) -> F : A --> ( Base ` E ) )
33		frn 5719	. . . . . . . . . 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 14778	. . . . . . . . 9 |- ( Base ` E ) C_ ( Base ` D )
37	34, 36	syl6ss 3501	. . . . . . . 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 3708	. . . . . 6 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> ran F C_ ( S i^i ( Base ` D ) ) )
40		df-f 5574	. . . . . 6 |- ( F : A --> ( S i^i ( Base ` D ) ) <-> ( F Fn A /\ ran F C_ ( S i^i ( Base ` D ) ) ) )
41	23, 39, 40	sylanbrc 662	. . . . 5 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> F : A --> ( S i^i ( Base ` D ) ) )
42		eqid 2454	. . . . . . . . 9 |- ( Hom ` D ) = ( Hom ` D )
43		simpr 459	. . . . . . . . 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 15359	. . . . . . . 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 2454	. . . . . . . . . 10 |- ( Hom ` E ) = ( Hom ` E )
48		simpr 459	. . . . . . . . . 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 15359	. . . . . . . . 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		funcres2c.r	. . . . . . . . . . . . 13 |- ( ph -> S e. V )
53	35, 42	resshom 14910	. . . . . . . . . . . . 13 |- ( S e. V -> ( Hom ` D ) = ( Hom ` E ) )
54	52, 53	syl 16	. . . . . . . . . . . 12 |- ( ph -> ( Hom ` D ) = ( Hom ` E ) )
55	54	ad2antrr 723	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. A /\ y e. A ) ) /\ F ( C Func E ) G ) -> ( Hom ` D ) = ( Hom ` E ) )
56	55	oveqd 6287	. . . . . . . . . 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 ) ) )
57	56	feq3d 5701	. . . . . . . . 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 ) ) ) )
58	51, 57	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 ) ) )
59	46, 58	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 ) ) )
60	59	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 ) ) )
61	41	adantr 463	. . . . . . . . . 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 ) ) )
62		simprl 754	. . . . . . . . . 10 |- ( ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) /\ ( x e. A /\ y e. A ) ) -> x e. A )
63	61, 62	ffvelrnd 6008	. . . . . . . . 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 ) ) )
64		simprr 755	. . . . . . . . . 10 |- ( ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) /\ ( x e. A /\ y e. A ) ) -> y e. A )
65	61, 64	ffvelrnd 6008	. . . . . . . . 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 ) ) )
66	63, 65	ovresd 6416	. . . . . . . 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 ) ) )
67	10, 63	sseldi 3487	. . . . . . . . 9 |- ( ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) /\ ( x e. A /\ y e. A ) ) -> ( F ` x ) e. ( Base ` D ) )
68	10, 65	sseldi 3487	. . . . . . . . 9 |- ( ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) /\ ( x e. A /\ y e. A ) ) -> ( F ` y ) e. ( Base ` D ) )
69	8, 7, 42, 67, 68	homfval 15183	. . . . . . . 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 ) ) )
70	66, 69	eqtrd 2495	. . . . . . 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 ) ) )
71	70	feq3d 5701	. . . . . 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 ) ) ) )
72	60, 71	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 ) ) )
73	5, 6, 13, 19, 41, 72	funcres2b 15388	. . . 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 ) )
74		eqidd 2455	. . . . . 6 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> ( Hom f ` C ) = ( Hom f ` C ) )
75		eqidd 2455	. . . . . 6 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> (compf ` C ) = (compf ` C ) )
76	7	ressinbas 14782	. . . . . . . . . . 11 |- ( S e. V -> ( D ↾s S ) = ( D ↾s ( S i^i ( Base ` D ) ) ) )
77	52, 76	syl 16	. . . . . . . . . 10 |- ( ph -> ( D ↾s S ) = ( D ↾s ( S i^i ( Base ` D ) ) ) )
78	35, 77	syl5eq 2507	. . . . . . . . 9 |- ( ph -> E = ( D ↾s ( S i^i ( Base ` D ) ) ) )
79	78	fveq2d 5852	. . . . . . . 8 |- ( ph -> ( Hom f ` E ) = ( Hom f ` ( D ↾s ( S i^i ( Base ` D ) ) ) ) )
80		eqid 2454	. . . . . . . . . 10 |- ( D ↾s ( S i^i ( Base ` D ) ) ) = ( D ↾s ( S i^i ( Base ` D ) ) )
81		eqid 2454	. . . . . . . . . 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 ) ) ) ) )
82	7, 8, 9, 11, 80, 81	fullresc 15342	. . . . . . . . 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 ) ) ) ) ) ) ) )
83	82	simpld 457	. . . . . . . 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 ) ) ) ) ) ) )
84	79, 83	eqtrd 2495	. . . . . . 7 |- ( ph -> ( Hom f ` E ) = ( Hom f ` ( D |`cat ( ( Hom f ` D ) |` ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) ) ) ) )
85	84	adantr 463	. . . . . 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 ) ) ) ) ) ) )
86	78	fveq2d 5852	. . . . . . . 8 |- ( ph -> (compf ` E ) = (compf ` ( D ↾s ( S i^i ( Base ` D ) ) ) ) )
87	82	simprd 461	. . . . . . . 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 ) ) ) ) ) ) )
88	86, 87	eqtrd 2495	. . . . . . 7 |- ( ph -> (compf ` E ) = (compf ` ( D |`cat ( ( Hom f ` D ) |` ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) ) ) ) )
89	88	adantr 463	. . . . . 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 ) ) ) ) ) ) )
90		df-br 4440	. . . . . . . . . . 11 |- ( F ( C Func D ) G <-> <. F , G >. e. ( C Func D ) )
91		funcrcl 15354	. . . . . . . . . . 11 |- ( <. F , G >. e. ( C Func D ) -> ( C e. Cat /\ D e. Cat ) )
92	90, 91	sylbi 195	. . . . . . . . . 10 |- ( F ( C Func D ) G -> ( C e. Cat /\ D e. Cat ) )
93	92	simpld 457	. . . . . . . . 9 |- ( F ( C Func D ) G -> C e. Cat )
94		df-br 4440	. . . . . . . . . . 11 |- ( F ( C Func E ) G <-> <. F , G >. e. ( C Func E ) )
95		funcrcl 15354	. . . . . . . . . . 11 |- ( <. F , G >. e. ( C Func E ) -> ( C e. Cat /\ E e. Cat ) )
96	94, 95	sylbi 195	. . . . . . . . . 10 |- ( F ( C Func E ) G -> ( C e. Cat /\ E e. Cat ) )
97	96	simpld 457	. . . . . . . . 9 |- ( F ( C Func E ) G -> C e. Cat )
98	93, 97	jaoi 377	. . . . . . . 8 |- ( ( F ( C Func D ) G \/ F ( C Func E ) G ) -> C e. Cat )
99		elex 3115	. . . . . . . 8 |- ( C e. Cat -> C e. _V )
100	98, 99	syl 16	. . . . . . 7 |- ( ( F ( C Func D ) G \/ F ( C Func E ) G ) -> C e. _V )
101	100	adantl 464	. . . . . 6 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> C e. _V )
102		ovex 6298	. . . . . . . 8 |- ( D ↾s S ) e. _V
103	35, 102	eqeltri 2538	. . . . . . 7 |- E e. _V
104	103	a1i 11	. . . . . 6 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> E e. _V )
105		ovex 6298	. . . . . . 7 |- ( D |`cat ( ( Hom f ` D ) |` ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) ) ) e. _V
106	105	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 )
107	74, 75, 85, 89, 101, 101, 104, 106	funcpropd 15391	. . . . 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 ) ) ) ) ) ) )
108	107	breqd 4450	. . . 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 ) )
109	73, 108	bitr4d 256	. . 3 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> ( F ( C Func D ) G <-> F ( C Func E ) G ) )
110	109	ex 432	. 2 |- ( ph -> ( ( F ( C Func D ) G \/ F ( C Func E ) G ) -> ( F ( C Func D ) G <-> F ( C Func E ) G ) ) )
111	2, 4, 110	pm5.21ndd 352	1 |- ( ph -> ( F ( C Func D ) G <-> F ( C Func E ) G ) )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 366   /\ wa 367   = wceq 1398   e. wcel 1823   _Vcvv 3106   i^i cin 3460   C_ wss 3461   <.cop 4022   class class class wbr 4439   X. cxp 4986   ran crn 4989   |` cres 4990   Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270   Basecbs 14719   ↾_s cress 14720   Hom chom 14798   Catccat 15156   Hom _f chomf 15158  comp_fccomf 15159   |`_cat cresc 15299  Subcatcsubc 15300   Func cfunc 15345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-hom 14811  df-cco 14812  df-cat 15160  df-cid 15161  df-homf 15162  df-comf 15163  df-ssc 15301  df-resc 15302  df-subc 15303  df-func 15349

This theorem is referenced by:  fthres2c  15422  fullres2c  15430