Theorem funcres2c 15139

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 2441	. . . . 5 |- ( Hom ` C ) = ( Hom ` C )
7		eqid 2441	. . . . . . 7 |- ( Base ` D ) = ( Base ` D )
8		eqid 2441	. . . . . . 7 |- ( Hom f ` D ) = ( Hom f ` D )
9		funcres2c.d	. . . . . . 7 |- ( ph -> D e. Cat )
10		inss2 3701	. . . . . . . 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 15088	. . . . . 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 14960	. . . . . . 7 |- ( Hom f ` D ) Fn ( ( Base ` D ) X. ( Base ` D ) )
15		xpss12 5094	. . . . . . . 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 5680	. . . . . . 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 5717	. . . . . . 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 5723	. . . . . . . 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 15104	. . . . . . . . 9 |- ( ( ph /\ F ( C Func D ) G ) -> F : A --> ( Base ` D ) )
28		frn 5723	. . . . . . . . 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 2441	. . . . . . . . . . 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 15104	. . . . . . . . . 10 |- ( ( ph /\ F ( C Func E ) G ) -> F : A --> ( Base ` E ) )
33		frn 5723	. . . . . . . . . 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 14561	. . . . . . . . 9 |- ( Base ` E ) C_ ( Base ` D )
37	34, 36	syl6ss 3498	. . . . . . . 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 3704	. . . . . 6 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> ran F C_ ( S i^i ( Base ` D ) ) )
40		df-f 5578	. . . . . 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 2441	. . . . . . . . 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 15106	. . . . . . . 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 2441	. . . . . . . . . 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 15106	. . . . . . . . 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 14688	. . . . . . . . . . . . 13 |- ( S e. V -> ( Hom ` D ) = ( Hom ` E ) )
54	52, 53	syl 16	. . . . . . . . . . . 12 |- ( ph -> ( Hom ` D ) = ( Hom ` E ) )
55	54	ad2antrr 725	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. A /\ y e. A ) ) /\ F ( C Func E ) G ) -> ( Hom ` D ) = ( Hom ` E ) )
56	55	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 ) ) )
57	56	feq3d 5705	. . . . . . . . 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 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 ) ) )
62		simprl 755	. . . . . . . . . 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 6013	. . . . . . . . 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 756	. . . . . . . . . 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 6013	. . . . . . . . 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 6424	. . . . . . . 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 3484	. . . . . . . . 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 3484	. . . . . . . . 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 14959	. . . . . . . 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 2482	. . . . . . 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 5705	. . . . . 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 15135	. . . 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 2442	. . . . . 6 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> ( Hom f ` C ) = ( Hom f ` C ) )
75		eqidd 2442	. . . . . 6 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> (compf ` C ) = (compf ` C ) )
76	7	ressinbas 14565	. . . . . . . . . . 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 2494	. . . . . . . . 9 |- ( ph -> E = ( D ↾s ( S i^i ( Base ` D ) ) ) )
79	78	fveq2d 5856	. . . . . . . 8 |- ( ph -> ( Hom f ` E ) = ( Hom f ` ( D ↾s ( S i^i ( Base ` D ) ) ) ) )
80		eqid 2441	. . . . . . . . . 10 |- ( D ↾s ( S i^i ( Base ` D ) ) ) = ( D ↾s ( S i^i ( Base ` D ) ) )
81		eqid 2441	. . . . . . . . . 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 15089	. . . . . . . . 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 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 ) ) ) ) ) ) )
84	79, 83	eqtrd 2482	. . . . . . 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 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 ) ) ) ) ) ) )
86	78	fveq2d 5856	. . . . . . . 8 |- ( ph -> (compf ` E ) = (compf ` ( D ↾s ( S i^i ( Base ` D ) ) ) ) )
87	82	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 ) ) ) ) ) ) )
88	86, 87	eqtrd 2482	. . . . . . 7 |- ( ph -> (compf ` E ) = (compf ` ( D |`cat ( ( Hom f ` D ) |` ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) ) ) ) )
89	88	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 ) ) ) ) ) ) )
90		df-br 4434	. . . . . . . . . . 11 |- ( F ( C Func D ) G <-> <. F , G >. e. ( C Func D ) )
91		funcrcl 15101	. . . . . . . . . . 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 459	. . . . . . . . 9 |- ( F ( C Func D ) G -> C e. Cat )
94		df-br 4434	. . . . . . . . . . 11 |- ( F ( C Func E ) G <-> <. F , G >. e. ( C Func E ) )
95		funcrcl 15101	. . . . . . . . . . 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 459	. . . . . . . . 9 |- ( F ( C Func E ) G -> C e. Cat )
98	93, 97	jaoi 379	. . . . . . . 8 |- ( ( F ( C Func D ) G \/ F ( C Func E ) G ) -> C e. Cat )
99		elex 3102	. . . . . . . 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 466	. . . . . 6 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> C e. _V )
102		ovex 6305	. . . . . . . 8 |- ( D ↾s S ) e. _V
103	35, 102	eqeltri 2525	. . . . . . 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 6305	. . . . . . 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 15138	. . . . 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 4444	. . . 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 434	. 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 354	1 |- ( ph -> ( F ( C Func D ) G <-> F ( C Func E ) G ) )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1381   e. wcel 1802   _Vcvv 3093   i^i cin 3457   C_ wss 3458   <.cop 4016   class class class wbr 4433   X. cxp 4983   ran crn 4986   |` cres 4987   Fn wfn 5569   -->wf 5570   ` cfv 5574  (class class class)co 6277   Basecbs 14504   ↾_s cress 14505   Hom chom 14580   Catccat 14933   Hom _f chomf 14935  comp_fccomf 14936   |`_cat cresc 15049  Subcatcsubc 15050   Func cfunc 15092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-er 7309  df-map 7420  df-pm 7421  df-ixp 7468  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10980  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-hom 14593  df-cco 14594  df-cat 14937  df-cid 14938  df-homf 14939  df-comf 14940  df-ssc 15051  df-resc 15052  df-subc 15053  df-func 15096

This theorem is referenced by:  fthres2c  15169  fullres2c  15177