Theorem funcres2c 15884

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 392	. . 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 391	. . 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 2471	. . . . 5 |- ( Hom ` C ) = ( Hom ` C )
7		eqid 2471	. . . . . . 7 |- ( Base ` D ) = ( Base ` D )
8		eqid 2471	. . . . . . 7 |- ( Hom f ` D ) = ( Hom f ` D )
9		funcres2c.d	. . . . . . 7 |- ( ph -> D e. Cat )
10		inss2 3644	. . . . . . . 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 15833	. . . . . 6 |- ( ph -> ( ( Hom f ` D ) |` ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) ) e. (Subcat ` D ) )
13	12	adantr 472	. . . . 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 15676	. . . . . . 7 |- ( Hom f ` D ) Fn ( ( Base ` D ) X. ( Base ` D ) )
15		xpss12 4945	. . . . . . . 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 686	. . . . . . 7 |- ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) C_ ( ( Base ` D ) X. ( Base ` D ) )
17		fnssres 5699	. . . . . . 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 686	. . . . . 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 472	. . . . . . 7 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> F : A --> S )
22		ffn 5739	. . . . . . 7 |- ( F : A --> S -> F Fn A )
23	21, 22	syl 17	. . . . . 6 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> F Fn A )
24		frn 5747	. . . . . . . 8 |- ( F : A --> S -> ran F C_ S )
25	21, 24	syl 17	. . . . . . 7 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> ran F C_ S )
26		simpr 468	. . . . . . . . . 10 |- ( ( ph /\ F ( C Func D ) G ) -> F ( C Func D ) G )
27	5, 7, 26	funcf1 15849	. . . . . . . . 9 |- ( ( ph /\ F ( C Func D ) G ) -> F : A --> ( Base ` D ) )
28		frn 5747	. . . . . . . . 9 |- ( F : A --> ( Base ` D ) -> ran F C_ ( Base ` D ) )
29	27, 28	syl 17	. . . . . . . 8 |- ( ( ph /\ F ( C Func D ) G ) -> ran F C_ ( Base ` D ) )
30		eqid 2471	. . . . . . . . . . 11 |- ( Base ` E ) = ( Base ` E )
31		simpr 468	. . . . . . . . . . 11 |- ( ( ph /\ F ( C Func E ) G ) -> F ( C Func E ) G )
32	5, 30, 31	funcf1 15849	. . . . . . . . . 10 |- ( ( ph /\ F ( C Func E ) G ) -> F : A --> ( Base ` E ) )
33		frn 5747	. . . . . . . . . 10 |- ( F : A --> ( Base ` E ) -> ran F C_ ( Base ` E ) )
34	32, 33	syl 17	. . . . . . . . 9 |- ( ( ph /\ F ( C Func E ) G ) -> ran F C_ ( Base ` E ) )
35		funcres2c.e	. . . . . . . . . 10 |- E = ( D ↾s S )
36	35, 7	ressbasss 15259	. . . . . . . . 9 |- ( Base ` E ) C_ ( Base ` D )
37	34, 36	syl6ss 3430	. . . . . . . 8 |- ( ( ph /\ F ( C Func E ) G ) -> ran F C_ ( Base ` D ) )
38	29, 37	jaodan 802	. . . . . . 7 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> ran F C_ ( Base ` D ) )
39	25, 38	ssind 3647	. . . . . 6 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> ran F C_ ( S i^i ( Base ` D ) ) )
40		df-f 5593	. . . . . 6 |- ( F : A --> ( S i^i ( Base ` D ) ) <-> ( F Fn A /\ ran F C_ ( S i^i ( Base ` D ) ) ) )
41	23, 39, 40	sylanbrc 677	. . . . 5 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> F : A --> ( S i^i ( Base ` D ) ) )
42		eqid 2471	. . . . . . . . 9 |- ( Hom ` D ) = ( Hom ` D )
43		simpr 468	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. A /\ y e. A ) ) /\ F ( C Func D ) G ) -> F ( C Func D ) G )
44		simplrl 778	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. A /\ y e. A ) ) /\ F ( C Func D ) G ) -> x e. A )
45		simplrr 779	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. A /\ y e. A ) ) /\ F ( C Func D ) G ) -> y e. A )
46	5, 6, 42, 43, 44, 45	funcf2 15851	. . . . . . . 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 2471	. . . . . . . . . 10 |- ( Hom ` E ) = ( Hom ` E )
48		simpr 468	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. A /\ y e. A ) ) /\ F ( C Func E ) G ) -> F ( C Func E ) G )
49		simplrl 778	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. A /\ y e. A ) ) /\ F ( C Func E ) G ) -> x e. A )
50		simplrr 779	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. A /\ y e. A ) ) /\ F ( C Func E ) G ) -> y e. A )
51	5, 6, 47, 48, 49, 50	funcf2 15851	. . . . . . . . 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 15394	. . . . . . . . . . . . 13 |- ( S e. V -> ( Hom ` D ) = ( Hom ` E ) )
54	52, 53	syl 17	. . . . . . . . . . . 12 |- ( ph -> ( Hom ` D ) = ( Hom ` E ) )
55	54	ad2antrr 740	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. A /\ y e. A ) ) /\ F ( C Func E ) G ) -> ( Hom ` D ) = ( Hom ` E ) )
56	55	oveqd 6325	. . . . . . . . . 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 5726	. . . . . . . . 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 240	. . . . . . . 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 802	. . . . . . 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 821	. . . . . 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 472	. . . . . . . . . 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 772	. . . . . . . . . 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 6038	. . . . . . . . 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 774	. . . . . . . . . 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 6038	. . . . . . . . 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 6456	. . . . . . . 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 3416	. . . . . . . . 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 3416	. . . . . . . . 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 15675	. . . . . . . 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 2505	. . . . . . 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 5726	. . . . . 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 240	. . . . 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 15880	. . . 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 2472	. . . . . 6 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> ( Hom f ` C ) = ( Hom f ` C ) )
75		eqidd 2472	. . . . . 6 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> (compf ` C ) = (compf ` C ) )
76	7	ressinbas 15263	. . . . . . . . . . 11 |- ( S e. V -> ( D ↾s S ) = ( D ↾s ( S i^i ( Base ` D ) ) ) )
77	52, 76	syl 17	. . . . . . . . . 10 |- ( ph -> ( D ↾s S ) = ( D ↾s ( S i^i ( Base ` D ) ) ) )
78	35, 77	syl5eq 2517	. . . . . . . . 9 |- ( ph -> E = ( D ↾s ( S i^i ( Base ` D ) ) ) )
79	78	fveq2d 5883	. . . . . . . 8 |- ( ph -> ( Hom f ` E ) = ( Hom f ` ( D ↾s ( S i^i ( Base ` D ) ) ) ) )
80		eqid 2471	. . . . . . . . . 10 |- ( D ↾s ( S i^i ( Base ` D ) ) ) = ( D ↾s ( S i^i ( Base ` D ) ) )
81		eqid 2471	. . . . . . . . . 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 15834	. . . . . . . . 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 466	. . . . . . . 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 2505	. . . . . . 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 472	. . . . . 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 5883	. . . . . . . 8 |- ( ph -> (compf ` E ) = (compf ` ( D ↾s ( S i^i ( Base ` D ) ) ) ) )
87	82	simprd 470	. . . . . . . 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 2505	. . . . . . 7 |- ( ph -> (compf ` E ) = (compf ` ( D |`cat ( ( Hom f ` D ) |` ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) ) ) ) )
89	88	adantr 472	. . . . . 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 4396	. . . . . . . . . . 11 |- ( F ( C Func D ) G <-> <. F , G >. e. ( C Func D ) )
91		funcrcl 15846	. . . . . . . . . . 11 |- ( <. F , G >. e. ( C Func D ) -> ( C e. Cat /\ D e. Cat ) )
92	90, 91	sylbi 200	. . . . . . . . . 10 |- ( F ( C Func D ) G -> ( C e. Cat /\ D e. Cat ) )
93	92	simpld 466	. . . . . . . . 9 |- ( F ( C Func D ) G -> C e. Cat )
94		df-br 4396	. . . . . . . . . . 11 |- ( F ( C Func E ) G <-> <. F , G >. e. ( C Func E ) )
95		funcrcl 15846	. . . . . . . . . . 11 |- ( <. F , G >. e. ( C Func E ) -> ( C e. Cat /\ E e. Cat ) )
96	94, 95	sylbi 200	. . . . . . . . . 10 |- ( F ( C Func E ) G -> ( C e. Cat /\ E e. Cat ) )
97	96	simpld 466	. . . . . . . . 9 |- ( F ( C Func E ) G -> C e. Cat )
98	93, 97	jaoi 386	. . . . . . . 8 |- ( ( F ( C Func D ) G \/ F ( C Func E ) G ) -> C e. Cat )
99		elex 3040	. . . . . . . 8 |- ( C e. Cat -> C e. _V )
100	98, 99	syl 17	. . . . . . 7 |- ( ( F ( C Func D ) G \/ F ( C Func E ) G ) -> C e. _V )
101	100	adantl 473	. . . . . 6 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> C e. _V )
102		ovex 6336	. . . . . . . 8 |- ( D ↾s S ) e. _V
103	35, 102	eqeltri 2545	. . . . . . 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 6336	. . . . . . 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 15883	. . . . 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 4406	. . . 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 264	. . 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 441	. 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 361	1 |- ( ph -> ( F ( C Func D ) G <-> F ( C Func E ) G ) )

Syntax hints:   -> wi 4   <-> wb 189   \/ wo 375   /\ wa 376   = wceq 1452   e. wcel 1904   _Vcvv 3031   i^i cin 3389   C_ wss 3390   <.cop 3965   class class class wbr 4395   X. cxp 4837   ran crn 4840   |` cres 4841   Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   Basecbs 15199   ↾_s cress 15200   Hom chom 15279   Catccat 15648   Hom _f chomf 15650  comp_fccomf 15651   |`_cat cresc 15791  Subcatcsubc 15792   Func cfunc 15837

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-hom 15292  df-cco 15293  df-cat 15652  df-cid 15653  df-homf 15654  df-comf 15655  df-ssc 15793  df-resc 15794  df-subc 15795  df-func 15841

This theorem is referenced by:  fthres2c  15914  fullres2c  15922