Theorem dualded 15132

Description: The dual of a deductive system is a deductive system.

Hypotheses

Ref	Expression
dualcat2.1	|- D = (dom` T)
dualcat2.2	|- C = (cod` T)
dualcat2.3	|- J = (id` T)
dualcat2.4	|- R = (o` T)

Assertion

Ref	Expression
dualded	|- (T e. Ded -> <.<.C, D>., <.J, {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}>.>. e. Ded )

Distinct variable groups:   x,C,y,z   x,D,y,z   x,R,y,z   x,T,y,z

Proof of Theorem dualded

Step	Hyp	Ref	Expression
1		dedalg 15090	. . . . 5 |- (T e. Ded -> T e. Alg )
2		dualcat2.1	. . . . . 6 |- D = (dom` T)
3		dualcat2.2	. . . . . 6 |- C = (cod` T)
4		dualcat2.3	. . . . . 6 |- J = (id` T)
5		dualcat2.4	. . . . . 6 |- R = (o` T)
6	2, 3, 4, 5	dualalg 15131	. . . . 5 |- (T e. Alg -> <.<.C, D>., <.J, {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}>.>. e. Alg )
7	1, 6	syl 12	. . . 4 |- (T e. Ded -> <.<.C, D>., <.J, {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}>.>. e. Alg )
8		eqid 1884	. . . . . . 7 |- dom J = dom J
9	8, 2, 4, 3	idosd 15091	. . . . . 6 |- ((T e. Ded /\ a e. dom J) -> ((D` (J` a)) = a /\ (C` (J` a)) = a))
10	9	ancomd 483	. . . . 5 |- ((T e. Ded /\ a e. dom J) -> ((C` (J` a)) = a /\ (D` (J` a)) = a))
11	10	r19.21aiva 2176	. . . 4 |- (T e. Ded -> A.a e. dom J((C` (J` a)) = a /\ (D` (J` a)) = a))
12		relcnv 4301	. . . . . . . . . . 11 |- Rel `'dom R
13		oprex 4907	. . . . . . . . . . . 12 |- (yRx) e. _V
14	13	dmoprabss6 14336	. . . . . . . . . . 11 |- (Rel `'dom R -> dom {<.<.x, y>., z>. | (<.x, y>. e. `'dom R /\ z = (yRx))} = `'dom R)
15	12, 14	ax-mp 7	. . . . . . . . . 10 |- dom {<.<.x, y>., z>. | (<.x, y>. e. `'dom R /\ z = (yRx))} = `'dom R
16	15	a1i 8	. . . . . . . . 9 |- ((T e. Ded /\ (f e. dom C /\ g e. dom C)) -> dom {<.<.x, y>., z>. | (<.x, y>. e. `'dom R /\ z = (yRx))} = `'dom R)
17		visset 2295	. . . . . . . . . . . . . 14 |- x e. _V
18		visset 2295	. . . . . . . . . . . . . 14 |- y e. _V
19	17, 18	opelcnv 4143	. . . . . . . . . . . . 13 |- (<.x, y>. e. `'dom R <-> <.y, x>. e. dom R)
20	19	bicomi 189	. . . . . . . . . . . 12 |- (<.y, x>. e. dom R <-> <.x, y>. e. `'dom R)
21	20	anbi1i 539	. . . . . . . . . . 11 |- ((<.y, x>. e. dom R /\ z = (yRx)) <-> (<.x, y>. e. `'dom R /\ z = (yRx)))
22	21	oprabbii 4923	. . . . . . . . . 10 |- {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} = {<.<.x, y>., z>. | (<.x, y>. e. `'dom R /\ z = (yRx))}
23	22	dmeqi 4158	. . . . . . . . 9 |- dom {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} = dom {<.<.x, y>., z>. | (<.x, y>. e. `'dom R /\ z = (yRx))}
24	16, 23	syl5eq 1940	. . . . . . . 8 |- ((T e. Ded /\ (f e. dom C /\ g e. dom C)) -> dom {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} = `'dom R)
25	24	eleq2d 1964	. . . . . . 7 |- ((T e. Ded /\ (f e. dom C /\ g e. dom C)) -> (<.g, f>. e. dom {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} <-> <.g, f>. e. `'dom R))
26		visset 2295	. . . . . . . . . 10 |- g e. _V
27		visset 2295	. . . . . . . . . 10 |- f e. _V
28	26, 27	opelcnv 4143	. . . . . . . . 9 |- (<.g, f>. e. `'dom R <-> <.f, g>. e. dom R)
29	28	a1i 8	. . . . . . . 8 |- ((T e. Ded /\ (f e. dom C /\ g e. dom C)) -> (<.g, f>. e. `'dom R <-> <.f, g>. e. dom R))
30	2, 3	dcsda 15080	. . . . . . . . . . . . 13 |- (T e. Alg -> dom D = dom C)
31		eleq2 1958	. . . . . . . . . . . . . . . . 17 |- (dom C = dom D -> (g e. dom C <-> g e. dom D))
32		eleq2 1958	. . . . . . . . . . . . . . . . 17 |- (dom C = dom D -> (f e. dom C <-> f e. dom D))
33	31, 32	anbi12d 690	. . . . . . . . . . . . . . . 16 |- (dom C = dom D -> ((g e. dom C /\ f e. dom C) <-> (g e. dom D /\ f e. dom D)))
34	33	imbi1d 675	. . . . . . . . . . . . . . 15 |- (dom C = dom D -> (((g e. dom C /\ f e. dom C) -> (<.f, g>. e. dom R <-> (D` f) = (C` g))) <-> ((g e. dom D /\ f e. dom D) -> (<.f, g>. e. dom R <-> (D` f) = (C` g)))))
35		eqid 1884	. . . . . . . . . . . . . . . . 17 |- dom D = dom D
36	35, 2, 3, 5	cmppfd 15092	. . . . . . . . . . . . . . . 16 |- ((T e. Ded /\ g e. dom D /\ f e. dom D) -> (<.f, g>. e. dom R <-> (D` f) = (C` g)))
37	36	3expib 1070	. . . . . . . . . . . . . . 15 |- (T e. Ded -> ((g e. dom D /\ f e. dom D) -> (<.f, g>. e. dom R <-> (D` f) = (C` g))))
38	34, 37	syl5bir 227	. . . . . . . . . . . . . 14 |- (dom C = dom D -> (T e. Ded -> ((g e. dom C /\ f e. dom C) -> (<.f, g>. e. dom R <-> (D` f) = (C` g)))))
39	38	eqcoms 1887	. . . . . . . . . . . . 13 |- (dom D = dom C -> (T e. Ded -> ((g e. dom C /\ f e. dom C) -> (<.f, g>. e. dom R <-> (D` f) = (C` g)))))
40	1, 30, 39	3syl 24	. . . . . . . . . . . 12 |- (T e. Ded -> (T e. Ded -> ((g e. dom C /\ f e. dom C) -> (<.f, g>. e. dom R <-> (D` f) = (C` g)))))
41	40	pm2.43i 78	. . . . . . . . . . 11 |- (T e. Ded -> ((g e. dom C /\ f e. dom C) -> (<.f, g>. e. dom R <-> (D` f) = (C` g))))
42	41	com12 14	. . . . . . . . . 10 |- ((g e. dom C /\ f e. dom C) -> (T e. Ded -> (<.f, g>. e. dom R <-> (D` f) = (C` g))))
43	42	ancoms 484	. . . . . . . . 9 |- ((f e. dom C /\ g e. dom C) -> (T e. Ded -> (<.f, g>. e. dom R <-> (D` f) = (C` g))))
44	43	impcom 378	. . . . . . . 8 |- ((T e. Ded /\ (f e. dom C /\ g e. dom C)) -> (<.f, g>. e. dom R <-> (D` f) = (C` g)))
45		eqcom 1886	. . . . . . . . 9 |- ((D` f) = (C` g) <-> (C` g) = (D` f))
46	45	a1i 8	. . . . . . . 8 |- ((T e. Ded /\ (f e. dom C /\ g e. dom C)) -> ((D` f) = (C` g) <-> (C` g) = (D` f)))
47	29, 44, 46	3bitrd 603	. . . . . . 7 |- ((T e. Ded /\ (f e. dom C /\ g e. dom C)) -> (<.g, f>. e. `'dom R <-> (C` g) = (D` f)))
48	25, 47	bitrd 587	. . . . . 6 |- ((T e. Ded /\ (f e. dom C /\ g e. dom C)) -> (<.g, f>. e. dom {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} <-> (C` g) = (D` f)))
49	48	ex 402	. . . . 5 |- (T e. Ded -> ((f e. dom C /\ g e. dom C) -> (<.g, f>. e. dom {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} <-> (C` g) = (D` f))))
50	49	r19.21aivv 2183	. . . 4 |- (T e. Ded -> A.f e. dom CA.g e. dom C(<.g, f>. e. dom {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} <-> (C` g) = (D` f)))
51	7, 11, 50	3jca 1050	. . 3 |- (T e. Ded -> (<.<.C, D>., <.J, {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}>.>. e. Alg /\ A.a e. dom J((C` (J` a)) = a /\ (D` (J` a)) = a) /\ A.f e. dom CA.g e. dom C(<.g, f>. e. dom {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} <-> (C` g) = (D` f))))
52		simpll 448	. . . . . . 7 |- (((T e. Ded /\ (f e. dom C /\ g e. dom C)) /\ (C` g) = (D` f)) -> T e. Ded )
53	30	eqcomd 1889	. . . . . . . . . . . . 13 |- (T e. Alg -> dom C = dom D)
54	53	eleq2d 1964	. . . . . . . . . . . 12 |- (T e. Alg -> (f e. dom C <-> f e. dom D))
55	53	eleq2d 1964	. . . . . . . . . . . 12 |- (T e. Alg -> (g e. dom C <-> g e. dom D))
56	54, 55	anbi12d 690	. . . . . . . . . . 11 |- (T e. Alg -> ((f e. dom C /\ g e. dom C) <-> (f e. dom D /\ g e. dom D)))
57	1, 56	syl 12	. . . . . . . . . 10 |- (T e. Ded -> ((f e. dom C /\ g e. dom C) <-> (f e. dom D /\ g e. dom D)))
58	57	biimpa 460	. . . . . . . . 9 |- ((T e. Ded /\ (f e. dom C /\ g e. dom C)) -> (f e. dom D /\ g e. dom D))
59	58	ancomd 483	. . . . . . . 8 |- ((T e. Ded /\ (f e. dom C /\ g e. dom C)) -> (g e. dom D /\ f e. dom D))
60	59	adantr 425	. . . . . . 7 |- (((T e. Ded /\ (f e. dom C /\ g e. dom C)) /\ (C` g) = (D` f)) -> (g e. dom D /\ f e. dom D))
61		eqcom 1886	. . . . . . . . 9 |- ((C` g) = (D` f) <-> (D` f) = (C` g))
62	61	biimpi 168	. . . . . . . 8 |- ((C` g) = (D` f) -> (D` f) = (C` g))
63	62	adantl 424	. . . . . . 7 |- (((T e. Ded /\ (f e. dom C /\ g e. dom C)) /\ (C` g) = (D` f)) -> (D` f) = (C` g))
64	2, 3, 5	dualded2lem 15130	. . . . . . . 8 |- ((T e. Ded /\ (g e. dom D /\ f e. dom D) /\ (D` f) = (C` g)) -> (g{<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}f) = (fRg))
65	64	fveq2d 4685	. . . . . . 7 |- ((T e. Ded /\ (g e. dom D /\ f e. dom D) /\ (D` f) = (C` g)) -> (C` (g{<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}f)) = (C` (fRg)))
66	52, 60, 63, 65	syl111anc 1100	. . . . . 6 |- (((T e. Ded /\ (f e. dom C /\ g e. dom C)) /\ (C` g) = (D` f)) -> (C` (g{<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}f)) = (C` (fRg)))
67	35, 2, 3, 5	codcmpd 15094	. . . . . . . . . . . . . . . 16 |- ((T e. Ded /\ g e. dom D /\ f e. dom D) -> ((D` f) = (C` g) -> (C` (fRg)) = (C` f)))
68	67	3exp 1066	. . . . . . . . . . . . . . 15 |- (T e. Ded -> (g e. dom D -> (f e. dom D -> ((D` f) = (C` g) -> (C` (fRg)) = (C` f)))))
69	32	imbi1d 675	. . . . . . . . . . . . . . . . . 18 |- (dom C = dom D -> ((f e. dom C -> ((D` f) = (C` g) -> (C` (fRg)) = (C` f))) <-> (f e. dom D -> ((D` f) = (C` g) -> (C` (fRg)) = (C` f)))))
70	31, 69	imbi12d 688	. . . . . . . . . . . . . . . . 17 |- (dom C = dom D -> ((g e. dom C -> (f e. dom C -> ((D` f) = (C` g) -> (C` (fRg)) = (C` f)))) <-> (g e. dom D -> (f e. dom D -> ((D` f) = (C` g) -> (C` (fRg)) = (C` f))))))
71	70	imbi2d 674	. . . . . . . . . . . . . . . 16 |- (dom C = dom D -> ((T e. Ded -> (g e. dom C -> (f e. dom C -> ((D` f) = (C` g) -> (C` (fRg)) = (C` f))))) <-> (T e. Ded -> (g e. dom D -> (f e. dom D -> ((D` f) = (C` g) -> (C` (fRg)) = (C` f)))))))
72	71	eqcoms 1887	. . . . . . . . . . . . . . 15 |- (dom D = dom C -> ((T e. Ded -> (g e. dom C -> (f e. dom C -> ((D` f) = (C` g) -> (C` (fRg)) = (C` f))))) <-> (T e. Ded -> (g e. dom D -> (f e. dom D -> ((D` f) = (C` g) -> (C` (fRg)) = (C` f)))))))
73	68, 72	mpbiri 211	. . . . . . . . . . . . . 14 |- (dom D = dom C -> (T e. Ded -> (g e. dom C -> (f e. dom C -> ((D` f) = (C` g) -> (C` (fRg)) = (C` f))))))
74	1, 30, 73	3syl 24	. . . . . . . . . . . . 13 |- (T e. Ded -> (T e. Ded -> (g e. dom C -> (f e. dom C -> ((D` f) = (C` g) -> (C` (fRg)) = (C` f))))))
75	74	pm2.43i 78	. . . . . . . . . . . 12 |- (T e. Ded -> (g e. dom C -> (f e. dom C -> ((D` f) = (C` g) -> (C` (fRg)) = (C` f)))))
76	75	com3l 38	. . . . . . . . . . 11 |- (g e. dom C -> (f e. dom C -> (T e. Ded -> ((D` f) = (C` g) -> (C` (fRg)) = (C` f)))))
77	76	impcom 378	. . . . . . . . . 10 |- ((f e. dom C /\ g e. dom C) -> (T e. Ded -> ((D` f) = (C` g) -> (C` (fRg)) = (C` f))))
78	77	impcom 378	. . . . . . . . 9 |- ((T e. Ded /\ (f e. dom C /\ g e. dom C)) -> ((D` f) = (C` g) -> (C` (fRg)) = (C` f)))
79	78	com12 14	. . . . . . . 8 |- ((D` f) = (C` g) -> ((T e. Ded /\ (f e. dom C /\ g e. dom C)) -> (C` (fRg)) = (C` f)))
80	79	eqcoms 1887	. . . . . . 7 |- ((C` g) = (D` f) -> ((T e. Ded /\ (f e. dom C /\ g e. dom C)) -> (C` (fRg)) = (C` f)))
81	80	impcom 378	. . . . . 6 |- (((T e. Ded /\ (f e. dom C /\ g e. dom C)) /\ (C` g) = (D` f)) -> (C` (fRg)) = (C` f))
82	66, 81	eqtrd 1925	. . . . 5 |- (((T e. Ded /\ (f e. dom C /\ g e. dom C)) /\ (C` g) = (D` f)) -> (C` (g{<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}f)) = (C` f))
83	82	exp31 407	. . . 4 |- (T e. Ded -> ((f e. dom C /\ g e. dom C) -> ((C` g) = (D` f) -> (C` (g{<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}f)) = (C` f))))
84	83	r19.21aivv 2183	. . 3 |- (T e. Ded -> A.f e. dom CA.g e. dom C((C` g) = (D` f) -> (C` (g{<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}f)) = (C` f)))
85	64	fveq2d 4685	. . . . . . 7 |- ((T e. Ded /\ (g e. dom D /\ f e. dom D) /\ (D` f) = (C` g)) -> (D` (g{<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}f)) = (D` (fRg)))
86	52, 60, 63, 85	syl111anc 1100	. . . . . 6 |- (((T e. Ded /\ (f e. dom C /\ g e. dom C)) /\ (C` g) = (D` f)) -> (D` (g{<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}f)) = (D` (fRg)))
87		id 73	. . . . . . . . . . . . . . . . . 18 |- (dom C = dom D -> dom C = dom D)
88	87	eqcoms 1887	. . . . . . . . . . . . . . . . 17 |- (dom D = dom C -> dom C = dom D)
89	88	eleq2d 1964	. . . . . . . . . . . . . . . 16 |- (dom D = dom C -> (g e. dom C <-> g e. dom D))
90	88	eleq2d 1964	. . . . . . . . . . . . . . . . 17 |- (dom D = dom C -> (f e. dom C <-> f e. dom D))
91	90	imbi1d 675	. . . . . . . . . . . . . . . 16 |- (dom D = dom C -> ((f e. dom C -> ((D` f) = (C` g) -> (D` (fRg)) = (D` g))) <-> (f e. dom D -> ((D` f) = (C` g) -> (D` (fRg)) = (D` g)))))
92	89, 91	imbi12d 688	. . . . . . . . . . . . . . 15 |- (dom D = dom C -> ((g e. dom C -> (f e. dom C -> ((D` f) = (C` g) -> (D` (fRg)) = (D` g)))) <-> (g e. dom D -> (f e. dom D -> ((D` f) = (C` g) -> (D` (fRg)) = (D` g))))))
93	35, 2, 3, 5	domcmpd 15093	. . . . . . . . . . . . . . . 16 |- ((T e. Ded /\ g e. dom D /\ f e. dom D) -> ((D` f) = (C` g) -> (D` (fRg)) = (D` g)))
94	93	3exp 1066	. . . . . . . . . . . . . . 15 |- (T e. Ded -> (g e. dom D -> (f e. dom D -> ((D` f) = (C` g) -> (D` (fRg)) = (D` g)))))
95	92, 94	syl5bir 227	. . . . . . . . . . . . . 14 |- (dom D = dom C -> (T e. Ded -> (g e. dom C -> (f e. dom C -> ((D` f) = (C` g) -> (D` (fRg)) = (D` g))))))
96	1, 30, 95	3syl 24	. . . . . . . . . . . . 13 |- (T e. Ded -> (T e. Ded -> (g e. dom C -> (f e. dom C -> ((D` f) = (C` g) -> (D` (fRg)) = (D` g))))))
97	96	pm2.43i 78	. . . . . . . . . . . 12 |- (T e. Ded -> (g e. dom C -> (f e. dom C -> ((D` f) = (C` g) -> (D` (fRg)) = (D` g)))))
98	97	com3l 38	. . . . . . . . . . 11 |- (g e. dom C -> (f e. dom C -> (T e. Ded -> ((D` f) = (C` g) -> (D` (fRg)) = (D` g)))))
99	98	impcom 378	. . . . . . . . . 10 |- ((f e. dom C /\ g e. dom C) -> (T e. Ded -> ((D` f) = (C` g) -> (D` (fRg)) = (D` g))))
100	99	impcom 378	. . . . . . . . 9 |- ((T e. Ded /\ (f e. dom C /\ g e. dom C)) -> ((D` f) = (C` g) -> (D` (fRg)) = (D` g)))
101	100	com12 14	. . . . . . . 8 |- ((D` f) = (C` g) -> ((T e. Ded /\ (f e. dom C /\ g e. dom C)) -> (D` (fRg)) = (D` g)))
102	101	eqcoms 1887	. . . . . . 7 |- ((C` g) = (D` f) -> ((T e. Ded /\ (f e. dom C /\ g e. dom C)) -> (D` (fRg)) = (D` g)))
103	102	impcom 378	. . . . . 6 |- (((T e. Ded /\ (f e. dom C /\ g e. dom C)) /\ (C` g) = (D` f)) -> (D` (fRg)) = (D` g))
104	86, 103	eqtrd 1925	. . . . 5 |- (((T e. Ded /\ (f e. dom C /\ g e. dom C)) /\ (C` g) = (D` f)) -> (D` (g{<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}f)) = (D` g))
105	104	exp31 407	. . . 4 |- (T e. Ded -> ((f e. dom C /\ g e. dom C) -> ((C` g) = (D` f) -> (D` (g{<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}f)) = (D` g))))
106	105	r19.21aivv 2183	. . 3 |- (T e. Ded -> A.f e. dom CA.g e. dom C((C` g) = (D` f) -> (D` (g{<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}f)) = (D` g)))
107	51, 84, 106	jca32 312	. 2 |- (T e. Ded -> ((<.<.C, D>., <.J, {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}>.>. e. Alg /\ A.a e. dom J((C` (J` a)) = a /\ (D` (J` a)) = a) /\ A.f e. dom CA.g e. dom C(<.g, f>. e. dom {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} <-> (C` g) = (D` f))) /\ (A.f e. dom CA.g e. dom C((C` g) = (D` f) -> (C` (g{<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}f)) = (C` f)) /\ A.f e. dom CA.g e. dom C((C` g) = (D` f) -> (D` (g{<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}f)) = (D` g)))))
108		eqid 1884	. . . 4 |- dom C = dom C
109	108, 8	isded 15083	. . 3 |- (((C e. _V /\ D e. _V /\ J e. _V) /\ {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} e. _V) -> (<.<.C, D>., <.J, {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}>.>. e. Ded <-> ((<.<.C, D>., <.J, {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}>.>. e. Alg /\ A.a e. dom J((C` (J` a)) = a /\ (D` (J` a)) = a) /\ A.f e. dom CA.g e. dom C(<.g, f>. e. dom {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} <-> (C` g) = (D` f))) /\ (A.f e. dom CA.g e. dom C((C` g) = (D` f) -> (C` (g{<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}f)) = (C` f)) /\ A.f e. dom CA.g e. dom C((C` g) = (D` f) -> (D` (g{<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}f)) = (D` g))))))
110	3	eleq1i 1960	. . . . 5 |- (C e. _V <-> (cod` T) e. _V)
111	2	eleq1i 1960	. . . . 5 |- (D e. _V <-> (dom` T) e. _V)
112	4	eleq1i 1960	. . . . 5 |- (J e. _V <-> (id` T) e. _V)
113	110, 111, 112	3anbi123i 1056	. . . 4 |- ((C e. _V /\ D e. _V /\ J e. _V) <-> ((cod` T) e. _V /\ (dom` T) e. _V /\ (id` T) e. _V))
114		fvex 4689	. . . 4 |- (cod` T) e. _V
115		fvex 4689	. . . 4 |- (dom` T) e. _V
116		fvex 4689	. . . 4 |- (id` T) e. _V
117	113, 114, 115, 116	mpbir3an 1052	. . 3 |- (C e. _V /\ D e. _V /\ J e. _V)
118		ssexg 3457	. . . 4 |- (({<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} C_ {<.<.x, y>., z>. | (<.y, x>. e. (dom D X. dom D) /\ z = (yRx))} /\ {<.<.x, y>., z>. | (<.y, x>. e. (dom D X. dom D) /\ z = (yRx))} e. _V) -> {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} e. _V)
119	35, 2, 5	cmppfa 15079	. . . . . . . . 9 |- (T e. Alg -> (Fun R /\ dom R C_ (dom D X. dom D) /\ ran R C_ dom D))
120	1, 119	syl 12	. . . . . . . 8 |- (T e. Ded -> (Fun R /\ dom R C_ (dom D X. dom D) /\ ran R C_ dom D))
121	120	simp2d 889	. . . . . . 7 |- (T e. Ded -> dom R C_ (dom D X. dom D))
122	121	sseld 2619	. . . . . 6 |- (T e. Ded -> (<.y, x>. e. dom R -> <.y, x>. e. (dom D X. dom D)))
123	122	anim1d 619	. . . . 5 |- (T e. Ded -> ((<.y, x>. e. dom R /\ z = (yRx)) -> (<.y, x>. e. (dom D X. dom D) /\ z = (yRx))))
124	123	ssoprab2g 14333	. . . 4 |- (T e. Ded -> {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} C_ {<.<.x, y>., z>. | (<.y, x>. e. (dom D X. dom D) /\ z = (yRx))})
125	17, 18	opelcnv 4143	. . . . . . . 8 |- (<.x, y>. e. `'(dom D X. dom D) <-> <.y, x>. e. (dom D X. dom D))
126		cnvxp 4332	. . . . . . . . 9 |- `'(dom D X. dom D) = (dom D X. dom D)
127	126	eleq2i 1961	. . . . . . . 8 |- (<.x, y>. e. `'(dom D X. dom D) <-> <.x, y>. e. (dom D X. dom D))
128	125, 127	bitr3i 192	. . . . . . 7 |- (<.y, x>. e. (dom D X. dom D) <-> <.x, y>. e. (dom D X. dom D))
129	128	anbi1i 539	. . . . . 6 |- ((<.y, x>. e. (dom D X. dom D) /\ z = (yRx)) <-> (<.x, y>. e. (dom D X. dom D) /\ z = (yRx)))
130	129	oprabbii 4923	. . . . 5 |- {<.<.x, y>., z>. | (<.y, x>. e. (dom D X. dom D) /\ z = (yRx))} = {<.<.x, y>., z>. | (<.x, y>. e. (dom D X. dom D) /\ z = (yRx))}
131	2, 115	eqeltri 1967	. . . . . . . 8 |- D e. _V
132	131	dmex 4208	. . . . . . 7 |- dom D e. _V
133	132, 132	xpex 4096	. . . . . 6 |- (dom D X. dom D) e. _V
134		relxp 4088	. . . . . 6 |- Rel (dom D X. dom D)
135		oprabex2gpop 14337	. . . . . 6 |- (((dom D X. dom D) e. _V /\ Rel (dom D X. dom D)) -> {<.<.x, y>., z>. | (<.x, y>. e. (dom D X. dom D) /\ z = (yRx))} e. _V)
136	133, 134, 135	mp2an 761	. . . . 5 |- {<.<.x, y>., z>. | (<.x, y>. e. (dom D X. dom D) /\ z = (yRx))} e. _V
137	130, 136	eqeltri 1967	. . . 4 |- {<.<.x, y>., z>. | (<.y, x>. e. (dom D X. dom D) /\ z = (yRx))} e. _V
138	118, 124, 137	sylancl 525	. . 3 |- (T e. Ded -> {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} e. _V)
139	109, 117, 138	sylancr 526	. 2 |- (T e. Ded -> (<.<.C, D>., <.J, {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}>.>. e. Ded <-> ((<.<.C, D>., <.J, {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}>.>. e. Alg /\ A.a e. dom J((C` (J` a)) = a /\ (D` (J` a)) = a) /\ A.f e. dom CA.g e. dom C(<.g, f>. e. dom {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} <-> (C` g) = (D` f))) /\ (A.f e. dom CA.g e. dom C((C` g) = (D` f) -> (C` (g{<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}f)) = (C` f)) /\ A.f e. dom CA.g e. dom C((C` g) = (D` f) -> (D` (g{<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}f)) = (D` g))))))
140	107, 139	mpbird 213	1 |- (T e. Ded -> <.<.C, D>., <.J, {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}>.>. e. Ded )

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292   C_ wss 2593  <.cop 3046   X. cxp 3984  `'ccnv 3985  dom cdm 3986  ran crn 3987  Rel wrel 3991  Fun wfun 3992  ` cfv 3998  (class class class)co 4884  {copab2 4885   Alg calg 15058  domcdom_ 15059  codccod_ 15060  idcid_ 15061  oco_ 15062   Ded cded 15081

This theorem is referenced by:  dualcat2 15133

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-int 3215  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fo 4012  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-alg 15063  df-doma 15064  df-coda 15065  df-ida 15066  df-cmpa 15067  df-ded 15082

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