Theorem funcres2c 14807

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 3568	. . . . . . . 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 14756	. . . . . 6 |- ( ph -> ( ( Hom f ` D ) |` ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) ) e. (Subcat ` D ) )
13	12	adantr 462	. . . . 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 14628	. . . . . . 7 |- ( Hom f ` D ) Fn ( ( Base ` D ) X. ( Base ` D ) )
15		xpss12 4941	. . . . . . . 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 667	. . . . . . 7 |- ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) C_ ( ( Base ` D ) X. ( Base ` D ) )
17		fnssres 5521	. . . . . . 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 667	. . . . . 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 462	. . . . . . 7 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> F : A --> S )
22		ffn 5556	. . . . . . 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 5562	. . . . . . . 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 458	. . . . . . . . . 10 |- ( ( ph /\ F ( C Func D ) G ) -> F ( C Func D ) G )
27	5, 7, 26	funcf1 14772	. . . . . . . . 9 |- ( ( ph /\ F ( C Func D ) G ) -> F : A --> ( Base ` D ) )
28		frn 5562	. . . . . . . . 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 458	. . . . . . . . . . 11 |- ( ( ph /\ F ( C Func E ) G ) -> F ( C Func E ) G )
32	5, 30, 31	funcf1 14772	. . . . . . . . . 10 |- ( ( ph /\ F ( C Func E ) G ) -> F : A --> ( Base ` E ) )
33		frn 5562	. . . . . . . . . 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 14226	. . . . . . . . 9 |- ( Base ` E ) C_ ( Base ` D )
37	34, 36	syl6ss 3365	. . . . . . . 8 |- ( ( ph /\ F ( C Func E ) G ) -> ran F C_ ( Base ` D ) )
38	29, 37	jaodan 778	. . . . . . 7 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> ran F C_ ( Base ` D ) )
39	25, 38	ssind 3571	. . . . . 6 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> ran F C_ ( S i^i ( Base ` D ) ) )
40		df-f 5419	. . . . . 6 |- ( F : A --> ( S i^i ( Base ` D ) ) <-> ( F Fn A /\ ran F C_ ( S i^i ( Base ` D ) ) ) )
41	23, 39, 40	sylanbrc 659	. . . . 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 458	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. A /\ y e. A ) ) /\ F ( C Func D ) G ) -> F ( C Func D ) G )
44		simplrl 754	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. A /\ y e. A ) ) /\ F ( C Func D ) G ) -> x e. A )
45		simplrr 755	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. A /\ y e. A ) ) /\ F ( C Func D ) G ) -> y e. A )
46	5, 6, 42, 43, 44, 45	funcf2 14774	. . . . . . . 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 458	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. A /\ y e. A ) ) /\ F ( C Func E ) G ) -> F ( C Func E ) G )
49		simplrl 754	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. A /\ y e. A ) ) /\ F ( C Func E ) G ) -> x e. A )
50		simplrr 755	. . . . . . . . . 10 |- ( ( ( ph /\ ( x e. A /\ y e. A ) ) /\ F ( C Func E ) G ) -> y e. A )
51	5, 6, 47, 48, 49, 50	funcf2 14774	. . . . . . . . 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 2442	. . . . . . . . . 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 14353	. . . . . . . . . . . . 13 |- ( S e. V -> ( Hom ` D ) = ( Hom ` E ) )
55	53, 54	syl 16	. . . . . . . . . . . 12 |- ( ph -> ( Hom ` D ) = ( Hom ` E ) )
56	55	ad2antrr 720	. . . . . . . . . . 11 |- ( ( ( ph /\ ( x e. A /\ y e. A ) ) /\ F ( C Func E ) G ) -> ( Hom ` D ) = ( Hom ` E ) )
57	56	oveqd 6107	. . . . . . . . . 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 5551	. . . . . . . . 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 778	. . . . . . 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 797	. . . . . 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 2442	. . . . . . 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 462	. . . . . . . . . 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 750	. . . . . . . . . 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 5841	. . . . . . . . 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 751	. . . . . . . . . 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 5841	. . . . . . . . 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 6230	. . . . . . . 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 3351	. . . . . . . . 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 3351	. . . . . . . . 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 14627	. . . . . . . 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 2473	. . . . . . 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 5551	. . . . . 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 14803	. . . 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 2442	. . . . . 6 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> ( Hom f ` C ) = ( Hom f ` C ) )
77		eqidd 2442	. . . . . 6 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> (compf ` C ) = (compf ` C ) )
78	7	ressinbas 14230	. . . . . . . . . . 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 2485	. . . . . . . . 9 |- ( ph -> E = ( D ↾s ( S i^i ( Base ` D ) ) ) )
81	80	fveq2d 5692	. . . . . . . 8 |- ( ph -> ( Hom f ` E ) = ( Hom f ` ( D ↾s ( S i^i ( Base ` D ) ) ) ) )
82		eqid 2441	. . . . . . . . . 10 |- ( D ↾s ( S i^i ( Base ` D ) ) ) = ( D ↾s ( S i^i ( Base ` D ) ) )
83		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 ) ) ) ) )
84	7, 8, 9, 11, 82, 83	fullresc 14757	. . . . . . . . 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 456	. . . . . . . 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 2473	. . . . . . 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 462	. . . . . 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 5692	. . . . . . . 8 |- ( ph -> (compf ` E ) = (compf ` ( D ↾s ( S i^i ( Base ` D ) ) ) ) )
89	84	simprd 460	. . . . . . . 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 2473	. . . . . . 7 |- ( ph -> (compf ` E ) = (compf ` ( D |`cat ( ( Hom f ` D ) |` ( ( S i^i ( Base ` D ) ) X. ( S i^i ( Base ` D ) ) ) ) ) ) )
91	90	adantr 462	. . . . . 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 4290	. . . . . . . . . . 11 |- ( F ( C Func D ) G <-> <. F , G >. e. ( C Func D ) )
93		funcrcl 14769	. . . . . . . . . . 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 456	. . . . . . . . 9 |- ( F ( C Func D ) G -> C e. Cat )
96		df-br 4290	. . . . . . . . . . 11 |- ( F ( C Func E ) G <-> <. F , G >. e. ( C Func E ) )
97		funcrcl 14769	. . . . . . . . . . 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 456	. . . . . . . . 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 2979	. . . . . . . 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 463	. . . . . 6 |- ( ( ph /\ ( F ( C Func D ) G \/ F ( C Func E ) G ) ) -> C e. _V )
104		ovex 6115	. . . . . . . 8 |- ( D ↾s S ) e. _V
105	35, 104	eqeltri 2511	. . . . . . 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 6115	. . . . . . 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 14806	. . . . 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 4300	. . . 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 1364   e. wcel 1761   _Vcvv 2970   i^i cin 3324   C_ wss 3325   <.cop 3880   class class class wbr 4289   X. cxp 4834   ran crn 4837   |` cres 4838   Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090   Basecbs 14170   ↾_s cress 14171   Hom chom 14245   Catccat 14598   Hom _f chomf 14600  comp_fccomf 14601   |`_cat cresc 14717  Subcatcsubc 14718   Func cfunc 14760

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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-hom 14258  df-cco 14259  df-cat 14602  df-cid 14603  df-homf 14604  df-comf 14605  df-ssc 14719  df-resc 14720  df-subc 14721  df-func 14764

This theorem is referenced by:  fthres2c  14837  fullres2c  14845