Theorem issubcat 15193

Description: The set of all the subcategories of T.

Hypotheses

Ref	Expression
issubcat.2	|- D = (dom` T)
issubcat.3	|- C = (cod` T)
issubcat.4	|- R = (o` T)
issubcat.5	|- J = (id` T)

Assertion

Ref	Expression
issubcat	|- (T e. Cat -> ( SubCat ` T) = {a e. Cat | ((id` a) C_ J /\ ((dom` a) C_ D /\ (cod` a) C_ C) /\ (o` a) C_ R)})

Distinct variable groups:   C,a   D,a   J,a   R,a   T,a

Proof of Theorem issubcat

Step	Hyp	Ref	Expression
1		fveq2 4681	. . . . . 6 |- (x = T -> (id` x) = (id` T))
2		issubcat.5	. . . . . 6 |- J = (id` T)
3	1, 2	syl6eqr 1946	. . . . 5 |- (x = T -> (id` x) = J)
4	3	sseq2d 2645	. . . 4 |- (x = T -> ((id` a) C_ (id` x) <-> (id` a) C_ J))
5		fveq2 4681	. . . . . . 7 |- (x = T -> (dom` x) = (dom` T))
6		issubcat.2	. . . . . . 7 |- D = (dom` T)
7	5, 6	syl6eqr 1946	. . . . . 6 |- (x = T -> (dom` x) = D)
8	7	sseq2d 2645	. . . . 5 |- (x = T -> ((dom` a) C_ (dom` x) <-> (dom` a) C_ D))
9		fveq2 4681	. . . . . . 7 |- (x = T -> (cod` x) = (cod` T))
10		issubcat.3	. . . . . . 7 |- C = (cod` T)
11	9, 10	syl6eqr 1946	. . . . . 6 |- (x = T -> (cod` x) = C)
12	11	sseq2d 2645	. . . . 5 |- (x = T -> ((cod` a) C_ (cod` x) <-> (cod` a) C_ C))
13	8, 12	anbi12d 690	. . . 4 |- (x = T -> (((dom` a) C_ (dom` x) /\ (cod` a) C_ (cod` x)) <-> ((dom` a) C_ D /\ (cod` a) C_ C)))
14		fveq2 4681	. . . . . 6 |- (x = T -> (o` x) = (o` T))
15		issubcat.4	. . . . . 6 |- R = (o` T)
16	14, 15	syl6eqr 1946	. . . . 5 |- (x = T -> (o` x) = R)
17	16	sseq2d 2645	. . . 4 |- (x = T -> ((o` a) C_ (o` x) <-> (o` a) C_ R))
18	4, 13, 17	3anbi123d 1168	. . 3 |- (x = T -> (((id` a) C_ (id` x) /\ ((dom` a) C_ (dom` x) /\ (cod` a) C_ (cod` x)) /\ (o` a) C_ (o` x)) <-> ((id` a) C_ J /\ ((dom` a) C_ D /\ (cod` a) C_ C) /\ (o` a) C_ R)))
19	18	rabbidv 2287	. 2 |- (x = T -> {a e. Cat | ((id` a) C_ (id` x) /\ ((dom` a) C_ (dom` x) /\ (cod` a) C_ (cod` x)) /\ (o` a) C_ (o` x))} = {a e. Cat | ((id` a) C_ J /\ ((dom` a) C_ D /\ (cod` a) C_ C) /\ (o` a) C_ R)})
20		df-subc 15192	. 2 |- SubCat = {<.x, y>. | (x e. Cat /\ y = {a e. Cat | ((id` a) C_ (id` x) /\ ((dom` a) C_ (dom` x) /\ (cod` a) C_ (cod` x)) /\ (o` a) C_ (o` x))})}
21		fvex 4689	. . . . . . 7 |- (dom` T) e. _V
22	6, 21	eqeltri 1967	. . . . . 6 |- D e. _V
23	22	pwex 3487	. . . . 5 |- ~PD e. _V
24		fvex 4689	. . . . . . 7 |- (cod` T) e. _V
25	10, 24	eqeltri 1967	. . . . . 6 |- C e. _V
26	25	pwex 3487	. . . . 5 |- ~PC e. _V
27	23, 26	xpex 4096	. . . 4 |- (~PD X. ~PC) e. _V
28		fvex 4689	. . . . . . 7 |- (id` T) e. _V
29	2, 28	eqeltri 1967	. . . . . 6 |- J e. _V
30	29	pwex 3487	. . . . 5 |- ~PJ e. _V
31		fvex 4689	. . . . . . 7 |- (o` T) e. _V
32	15, 31	eqeltri 1967	. . . . . 6 |- R e. _V
33	32	pwex 3487	. . . . 5 |- ~PR e. _V
34	30, 33	xpex 4096	. . . 4 |- (~PJ X. ~PR) e. _V
35	27, 34	xpex 4096	. . 3 |- ((~PD X. ~PC) X. (~PJ X. ~PR)) e. _V
36		relcat 15108	. . . . . . . 8 |- Rel Cat
37		1st2nd 5048	. . . . . . . 8 |- ((Rel Cat /\ u e. Cat ) -> u = <.(1st` u), (2nd` u)>.)
38	36, 37	mpan 759	. . . . . . 7 |- (u e. Cat -> u = <.(1st` u), (2nd` u)>.)
39	38	adantr 425	. . . . . 6 |- ((u e. Cat /\ ((id` u) C_ J /\ ((dom` u) C_ D /\ (cod` u) C_ C) /\ (o` u) C_ R)) -> u = <.(1st` u), (2nd` u)>.)
40		reldcat 15109	. . . . . . . . . 10 |- Rel dom Cat
41		1st2nd 5048	. . . . . . . . . . 11 |- ((Rel dom Cat /\ (1st` u) e. dom Cat ) -> (1st` u) = <.(1st` (1st` u)), (2nd` (1st` u))>.)
42		1stdm 5049	. . . . . . . . . . . 12 |- ((Rel Cat /\ u e. Cat ) -> (1st` u) e. dom Cat )
43	36, 42	mpan 759	. . . . . . . . . . 11 |- (u e. Cat -> (1st` u) e. dom Cat )
44	41, 43	sylan2 500	. . . . . . . . . 10 |- ((Rel dom Cat /\ u e. Cat ) -> (1st` u) = <.(1st` (1st` u)), (2nd` (1st` u))>.)
45	40, 44	mpan 759	. . . . . . . . 9 |- (u e. Cat -> (1st` u) = <.(1st` (1st` u)), (2nd` (1st` u))>.)
46	45	adantr 425	. . . . . . . 8 |- ((u e. Cat /\ ((id` u) C_ J /\ ((dom` u) C_ D /\ (cod` u) C_ C) /\ (o` u) C_ R)) -> (1st` u) = <.(1st` (1st` u)), (2nd` (1st` u))>.)
47		fvex 4689	. . . . . . . . . . . . . . . 16 |- (dom` u) e. _V
48	47	elpw 3037	. . . . . . . . . . . . . . 15 |- ((dom` u) e. ~PD <-> (dom` u) C_ D)
49		df-doma 15064	. . . . . . . . . . . . . . . . . . 19 |- dom = (1st o. 1st)
50	49	fveq1i 4682	. . . . . . . . . . . . . . . . . 18 |- (dom` u) = ((1st o. 1st)` u)
51		fo1st 5032	. . . . . . . . . . . . . . . . . . . 20 |- 1st:_V-onto->_V
52		fofun 4618	. . . . . . . . . . . . . . . . . . . 20 |- (1st:_V-onto->_V -> Fun 1st)
53	51, 52	ax-mp 7	. . . . . . . . . . . . . . . . . . 19 |- Fun 1st
54		visset 2295	. . . . . . . . . . . . . . . . . . . 20 |- u e. _V
55		fof 4617	. . . . . . . . . . . . . . . . . . . . . 22 |- (1st:_V-onto->_V -> 1st:_V-->_V)
56	51, 55	ax-mp 7	. . . . . . . . . . . . . . . . . . . . 21 |- 1st:_V-->_V
57	56	fdmi 4568	. . . . . . . . . . . . . . . . . . . 20 |- dom 1st = _V
58	54, 57	eleqtrri 1970	. . . . . . . . . . . . . . . . . . 19 |- u e. dom 1st
59		fvco 4736	. . . . . . . . . . . . . . . . . . 19 |- ((Fun 1st /\ Fun 1st /\ u e. dom 1st) -> ((1st o. 1st)` u) = (1st` (1st` u)))
60	53, 53, 58, 59	mp3an 1191	. . . . . . . . . . . . . . . . . 18 |- ((1st o. 1st)` u) = (1st` (1st` u))
61	50, 60	eqtri 1908	. . . . . . . . . . . . . . . . 17 |- (dom` u) = (1st` (1st` u))
62	61	eleq1i 1960	. . . . . . . . . . . . . . . 16 |- ((dom` u) e. ~PD <-> (1st` (1st` u)) e. ~PD)
63		ax-1 4	. . . . . . . . . . . . . . . 16 |- ((1st` (1st` u)) e. ~PD -> (u e. Cat -> (1st` (1st` u)) e. ~PD))
64	62, 63	sylbi 216	. . . . . . . . . . . . . . 15 |- ((dom` u) e. ~PD -> (u e. Cat -> (1st` (1st` u)) e. ~PD))
65	48, 64	sylbir 218	. . . . . . . . . . . . . 14 |- ((dom` u) C_ D -> (u e. Cat -> (1st` (1st` u)) e. ~PD))
66	65	a1d 15	. . . . . . . . . . . . 13 |- ((dom` u) C_ D -> ((o` u) C_ R -> (u e. Cat -> (1st` (1st` u)) e. ~PD)))
67	66	adantr 425	. . . . . . . . . . . 12 |- (((dom` u) C_ D /\ (cod` u) C_ C) -> ((o` u) C_ R -> (u e. Cat -> (1st` (1st` u)) e. ~PD)))
68	67	a1i 8	. . . . . . . . . . 11 |- ((id` u) C_ J -> (((dom` u) C_ D /\ (cod` u) C_ C) -> ((o` u) C_ R -> (u e. Cat -> (1st` (1st` u)) e. ~PD))))
69	68	3imp 1061	. . . . . . . . . 10 |- (((id` u) C_ J /\ ((dom` u) C_ D /\ (cod` u) C_ C) /\ (o` u) C_ R) -> (u e. Cat -> (1st` (1st` u)) e. ~PD))
70	69	impcom 378	. . . . . . . . 9 |- ((u e. Cat /\ ((id` u) C_ J /\ ((dom` u) C_ D /\ (cod` u) C_ C) /\ (o` u) C_ R)) -> (1st` (1st` u)) e. ~PD)
71		fvex 4689	. . . . . . . . . . . . . 14 |- (cod` u) e. _V
72	71	elpw 3037	. . . . . . . . . . . . 13 |- ((cod` u) e. ~PC <-> (cod` u) C_ C)
73		df-coda 15065	. . . . . . . . . . . . . . . . 17 |- cod = (2nd o. 1st)
74	73	fveq1i 4682	. . . . . . . . . . . . . . . 16 |- (cod` u) = ((2nd o. 1st)` u)
75		fo2nd 5033	. . . . . . . . . . . . . . . . . 18 |- 2nd:_V-onto->_V
76		fofun 4618	. . . . . . . . . . . . . . . . . 18 |- (2nd:_V-onto->_V -> Fun 2nd)
77	75, 76	ax-mp 7	. . . . . . . . . . . . . . . . 17 |- Fun 2nd
78		fvco 4736	. . . . . . . . . . . . . . . . 17 |- ((Fun 2nd /\ Fun 1st /\ u e. dom 1st) -> ((2nd o. 1st)` u) = (2nd` (1st` u)))
79	77, 53, 58, 78	mp3an 1191	. . . . . . . . . . . . . . . 16 |- ((2nd o. 1st)` u) = (2nd` (1st` u))
80	74, 79	eqtri 1908	. . . . . . . . . . . . . . 15 |- (cod` u) = (2nd` (1st` u))
81	80	eleq1i 1960	. . . . . . . . . . . . . 14 |- ((cod` u) e. ~PC <-> (2nd` (1st` u)) e. ~PC)
82		ax-1 4	. . . . . . . . . . . . . 14 |- ((2nd` (1st` u)) e. ~PC -> (u e. Cat -> (2nd` (1st` u)) e. ~PC))
83	81, 82	sylbi 216	. . . . . . . . . . . . 13 |- ((cod` u) e. ~PC -> (u e. Cat -> (2nd` (1st` u)) e. ~PC))
84	72, 83	sylbir 218	. . . . . . . . . . . 12 |- ((cod` u) C_ C -> (u e. Cat -> (2nd` (1st` u)) e. ~PC))
85	84	adantl 424	. . . . . . . . . . 11 |- (((dom` u) C_ D /\ (cod` u) C_ C) -> (u e. Cat -> (2nd` (1st` u)) e. ~PC))
86	85	3ad2ant2 898	. . . . . . . . . 10 |- (((id` u) C_ J /\ ((dom` u) C_ D /\ (cod` u) C_ C) /\ (o` u) C_ R) -> (u e. Cat -> (2nd` (1st` u)) e. ~PC))
87	86	impcom 378	. . . . . . . . 9 |- ((u e. Cat /\ ((id` u) C_ J /\ ((dom` u) C_ D /\ (cod` u) C_ C) /\ (o` u) C_ R)) -> (2nd` (1st` u)) e. ~PC)
88	70, 87	jca 310	. . . . . . . 8 |- ((u e. Cat /\ ((id` u) C_ J /\ ((dom` u) C_ D /\ (cod` u) C_ C) /\ (o` u) C_ R)) -> ((1st` (1st` u)) e. ~PD /\ (2nd` (1st` u)) e. ~PC))
89	46, 88	jca 310	. . . . . . 7 |- ((u e. Cat /\ ((id` u) C_ J /\ ((dom` u) C_ D /\ (cod` u) C_ C) /\ (o` u) C_ R)) -> ((1st` u) = <.(1st` (1st` u)), (2nd` (1st` u))>. /\ ((1st` (1st` u)) e. ~PD /\ (2nd` (1st` u)) e. ~PC)))
90		relrcat 15110	. . . . . . . . . . 11 |- Rel ran Cat
91		1st2nd 5048	. . . . . . . . . . . 12 |- ((Rel ran Cat /\ (2nd` u) e. ran Cat ) -> (2nd` u) = <.(1st` (2nd` u)), (2nd` (2nd` u))>.)
92		2ndrn 5050	. . . . . . . . . . . . 13 |- ((Rel Cat /\ u e. Cat ) -> (2nd` u) e. ran Cat )
93	36, 92	mpan 759	. . . . . . . . . . . 12 |- (u e. Cat -> (2nd` u) e. ran Cat )
94	91, 93	sylan2 500	. . . . . . . . . . 11 |- ((Rel ran Cat /\ u e. Cat ) -> (2nd` u) = <.(1st` (2nd` u)), (2nd` (2nd` u))>.)
95	90, 94	mpan 759	. . . . . . . . . 10 |- (u e. Cat -> (2nd` u) = <.(1st` (2nd` u)), (2nd` (2nd` u))>.)
96	95	adantr 425	. . . . . . . . 9 |- ((u e. Cat /\ ((id` u) C_ J /\ ((dom` u) C_ D /\ (cod` u) C_ C) /\ (o` u) C_ R)) -> (2nd` u) = <.(1st` (2nd` u)), (2nd` (2nd` u))>.)
97		fvex 4689	. . . . . . . . . . . . . 14 |- (id` u) e. _V
98	97	elpw 3037	. . . . . . . . . . . . 13 |- ((id` u) e. ~PJ <-> (id` u) C_ J)
99		df-ida 15066	. . . . . . . . . . . . . . . . 17 |- id = (1st o. 2nd)
100	99	fveq1i 4682	. . . . . . . . . . . . . . . 16 |- (id` u) = ((1st o. 2nd)` u)
101		fof 4617	. . . . . . . . . . . . . . . . . . . 20 |- (2nd:_V-onto->_V -> 2nd:_V-->_V)
102	75, 101	ax-mp 7	. . . . . . . . . . . . . . . . . . 19 |- 2nd:_V-->_V
103	102	fdmi 4568	. . . . . . . . . . . . . . . . . 18 |- dom 2nd = _V
104	54, 103	eleqtrri 1970	. . . . . . . . . . . . . . . . 17 |- u e. dom 2nd
105		fvco 4736	. . . . . . . . . . . . . . . . 17 |- ((Fun 1st /\ Fun 2nd /\ u e. dom 2nd) -> ((1st o. 2nd)` u) = (1st` (2nd` u)))
106	53, 77, 104, 105	mp3an 1191	. . . . . . . . . . . . . . . 16 |- ((1st o. 2nd)` u) = (1st` (2nd` u))
107	100, 106	eqtri 1908	. . . . . . . . . . . . . . 15 |- (id` u) = (1st` (2nd` u))
108	107	eleq1i 1960	. . . . . . . . . . . . . 14 |- ((id` u) e. ~PJ <-> (1st` (2nd` u)) e. ~PJ)
109		simp1 876	. . . . . . . . . . . . . . . 16 |- (((1st` (2nd` u)) e. ~PJ /\ (o` u) C_ R /\ u e. Cat ) -> (1st` (2nd` u)) e. ~PJ)
110		fvex 4689	. . . . . . . . . . . . . . . . . . . 20 |- (o` u) e. _V
111	110	elpw 3037	. . . . . . . . . . . . . . . . . . 19 |- ((o` u) e. ~PR <-> (o` u) C_ R)
112		df-cmpa 15067	. . . . . . . . . . . . . . . . . . . . . . 23 |- o = (2nd o. 2nd)
113	112	fveq1i 4682	. . . . . . . . . . . . . . . . . . . . . 22 |- (o` u) = ((2nd o. 2nd)` u)
114		fvco 4736	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((Fun 2nd /\ Fun 2nd /\ u e. dom 2nd) -> ((2nd o. 2nd)` u) = (2nd` (2nd` u)))
115	77, 77, 104, 114	mp3an 1191	. . . . . . . . . . . . . . . . . . . . . 22 |- ((2nd o. 2nd)` u) = (2nd` (2nd` u))
116	113, 115	eqtri 1908	. . . . . . . . . . . . . . . . . . . . 21 |- (o` u) = (2nd` (2nd` u))
117	116	eleq1i 1960	. . . . . . . . . . . . . . . . . . . 20 |- ((o` u) e. ~PR <-> (2nd` (2nd` u)) e. ~PR)
118		ax-1 4	. . . . . . . . . . . . . . . . . . . 20 |- ((2nd` (2nd` u)) e. ~PR -> (u e. Cat -> (2nd` (2nd` u)) e. ~PR))
119	117, 118	sylbi 216	. . . . . . . . . . . . . . . . . . 19 |- ((o` u) e. ~PR -> (u e. Cat -> (2nd` (2nd` u)) e. ~PR))
120	111, 119	sylbir 218	. . . . . . . . . . . . . . . . . 18 |- ((o` u) C_ R -> (u e. Cat -> (2nd` (2nd` u)) e. ~PR))
121	120	a1i 8	. . . . . . . . . . . . . . . . 17 |- ((1st` (2nd` u)) e. ~PJ -> ((o` u) C_ R -> (u e. Cat -> (2nd` (2nd` u)) e. ~PR)))
122	121	3imp 1061	. . . . . . . . . . . . . . . 16 |- (((1st` (2nd` u)) e. ~PJ /\ (o` u) C_ R /\ u e. Cat ) -> (2nd` (2nd` u)) e. ~PR)
123	109, 122	jca 310	. . . . . . . . . . . . . . 15 |- (((1st` (2nd` u)) e. ~PJ /\ (o` u) C_ R /\ u e. Cat ) -> ((1st` (2nd` u)) e. ~PJ /\ (2nd` (2nd` u)) e. ~PR))
124	123	3exp 1066	. . . . . . . . . . . . . 14 |- ((1st` (2nd` u)) e. ~PJ -> ((o` u) C_ R -> (u e. Cat -> ((1st` (2nd` u)) e. ~PJ /\ (2nd` (2nd` u)) e. ~PR))))
125	108, 124	sylbi 216	. . . . . . . . . . . . 13 |- ((id` u) e. ~PJ -> ((o` u) C_ R -> (u e. Cat -> ((1st` (2nd` u)) e. ~PJ /\ (2nd` (2nd` u)) e. ~PR))))
126	98, 125	sylbir 218	. . . . . . . . . . . 12 |- ((id` u) C_ J -> ((o` u) C_ R -> (u e. Cat -> ((1st` (2nd` u)) e. ~PJ /\ (2nd` (2nd` u)) e. ~PR))))
127	126	a1d 15	. . . . . . . . . . 11 |- ((id` u) C_ J -> (((dom` u) C_ D /\ (cod` u) C_ C) -> ((o` u) C_ R -> (u e. Cat -> ((1st` (2nd` u)) e. ~PJ /\ (2nd` (2nd` u)) e. ~PR)))))
128	127	3imp 1061	. . . . . . . . . 10 |- (((id` u) C_ J /\ ((dom` u) C_ D /\ (cod` u) C_ C) /\ (o` u) C_ R) -> (u e. Cat -> ((1st` (2nd` u)) e. ~PJ /\ (2nd` (2nd` u)) e. ~PR)))
129	128	impcom 378	. . . . . . . . 9 |- ((u e. Cat /\ ((id` u) C_ J /\ ((dom` u) C_ D /\ (cod` u) C_ C) /\ (o` u) C_ R)) -> ((1st` (2nd` u)) e. ~PJ /\ (2nd` (2nd` u)) e. ~PR))
130	96, 129	jca 310	. . . . . . . 8 |- ((u e. Cat /\ ((id` u) C_ J /\ ((dom` u) C_ D /\ (cod` u) C_ C) /\ (o` u) C_ R)) -> ((2nd` u) = <.(1st` (2nd` u)), (2nd` (2nd` u))>. /\ ((1st` (2nd` u)) e. ~PJ /\ (2nd` (2nd` u)) e. ~PR)))
131		elxp6 5041	. . . . . . . 8 |- ((2nd` u) e. (~PJ X. ~PR) <-> ((2nd` u) = <.(1st` (2nd` u)), (2nd` (2nd` u))>. /\ ((1st` (2nd` u)) e. ~PJ /\ (2nd` (2nd` u)) e. ~PR)))
132	130, 131	sylibr 217	. . . . . . 7 |- ((u e. Cat /\ ((id` u) C_ J /\ ((dom` u) C_ D /\ (cod` u) C_ C) /\ (o` u) C_ R)) -> (2nd` u) e. (~PJ X. ~PR))
133	89, 132	jca 310	. . . . . 6 |- ((u e. Cat /\ ((id` u) C_ J /\ ((dom` u) C_ D /\ (cod` u) C_ C) /\ (o` u) C_ R)) -> (((1st` u) = <.(1st` (1st` u)), (2nd` (1st` u))>. /\ ((1st` (1st` u)) e. ~PD /\ (2nd` (1st` u)) e. ~PC)) /\ (2nd` u) e. (~PJ X. ~PR)))
134	39, 133	jca 310	. . . . 5 |- ((u e. Cat /\ ((id` u) C_ J /\ ((dom` u) C_ D /\ (cod` u) C_ C) /\ (o` u) C_ R)) -> (u = <.(1st` u), (2nd` u)>. /\ (((1st` u) = <.(1st` (1st` u)), (2nd` (1st` u))>. /\ ((1st` (1st` u)) e. ~PD /\ (2nd` (1st` u)) e. ~PC)) /\ (2nd` u) e. (~PJ X. ~PR))))
135		fveq2 4681	. . . . . . . 8 |- (a = u -> (id` a) = (id` u))
136	135	sseq1d 2644	. . . . . . 7 |- (a = u -> ((id` a) C_ J <-> (id` u) C_ J))
137		fveq2 4681	. . . . . . . . 9 |- (a = u -> (dom` a) = (dom` u))
138	137	sseq1d 2644	. . . . . . . 8 |- (a = u -> ((dom` a) C_ D <-> (dom` u) C_ D))
139		fveq2 4681	. . . . . . . . 9 |- (a = u -> (cod` a) = (cod` u))
140	139	sseq1d 2644	. . . . . . . 8 |- (a = u -> ((cod` a) C_ C <-> (cod` u) C_ C))
141	138, 140	anbi12d 690	. . . . . . 7 |- (a = u -> (((dom` a) C_ D /\ (cod` a) C_ C) <-> ((dom` u) C_ D /\ (cod` u) C_ C)))
142		fveq2 4681	. . . . . . . 8 |- (a = u -> (o` a) = (o` u))
143	142	sseq1d 2644	. . . . . . 7 |- (a = u -> ((o` a) C_ R <-> (o` u) C_ R))
144	136, 141, 143	3anbi123d 1168	. . . . . 6 |- (a = u -> (((id` a) C_ J /\ ((dom` a) C_ D /\ (cod` a) C_ C) /\ (o` a) C_ R) <-> ((id` u) C_ J /\ ((dom` u) C_ D /\ (cod` u) C_ C) /\ (o` u) C_ R)))
145	144	elrab 2414	. . . . 5 |- (u e. {a e. Cat | ((id` a) C_ J /\ ((dom` a) C_ D /\ (cod` a) C_ C) /\ (o` a) C_ R)} <-> (u e. Cat /\ ((id` u) C_ J /\ ((dom` u) C_ D /\ (cod` u) C_ C) /\ (o` u) C_ R)))
146		elxp6 5041	. . . . . 6 |- (u e. ((~PD X. ~PC) X. (~PJ X. ~PR)) <-> (u = <.(1st` u), (2nd` u)>. /\ ((1st` u) e. (~PD X. ~PC) /\ (2nd` u) e. (~PJ X. ~PR))))
147		elxp6 5041	. . . . . . . 8 |- ((1st` u) e. (~PD X. ~PC) <-> ((1st` u) = <.(1st` (1st` u)), (2nd` (1st` u))>. /\ ((1st` (1st` u)) e. ~PD /\ (2nd` (1st` u)) e. ~PC)))
148	147	anbi1i 539	. . . . . . 7 |- (((1st` u) e. (~PD X. ~PC) /\ (2nd` u) e. (~PJ X. ~PR)) <-> (((1st` u) = <.(1st` (1st` u)), (2nd` (1st` u))>. /\ ((1st` (1st` u)) e. ~PD /\ (2nd` (1st` u)) e. ~PC)) /\ (2nd` u) e. (~PJ X. ~PR)))
149	148	anbi2i 538	. . . . . 6 |- ((u = <.(1st` u), (2nd` u)>. /\ ((1st` u) e. (~PD X. ~PC) /\ (2nd` u) e. (~PJ X. ~PR))) <-> (u = <.(1st` u), (2nd` u)>. /\ (((1st` u) = <.(1st` (1st` u)), (2nd` (1st` u))>. /\ ((1st` (1st` u)) e. ~PD /\ (2nd` (1st` u)) e. ~PC)) /\ (2nd` u) e. (~PJ X. ~PR))))
150	146, 149	bitri 190	. . . . 5 |- (u e. ((~PD X. ~PC) X. (~PJ X. ~PR)) <-> (u = <.(1st` u), (2nd` u)>. /\ (((1st` u) = <.(1st` (1st` u)), (2nd` (1st` u))>. /\ ((1st` (1st` u)) e. ~PD /\ (2nd` (1st` u)) e. ~PC)) /\ (2nd` u) e. (~PJ X. ~PR))))
151	134, 145, 150	3imtr4i 236	. . . 4 |- (u e. {a e. Cat | ((id` a) C_ J /\ ((dom` a) C_ D /\ (cod` a) C_ C) /\ (o` a) C_ R)} -> u e. ((~PD X. ~PC) X. (~PJ X. ~PR)))
152	151	ssriv 2621	. . 3 |- {a e. Cat | ((id` a) C_ J /\ ((dom` a) C_ D /\ (cod` a) C_ C) /\ (o` a) C_ R)} C_ ((~PD X. ~PC) X. (~PJ X. ~PR))
153	35, 152	ssexi 3456	. 2 |- {a e. Cat | ((id` a) C_ J /\ ((dom` a) C_ D /\ (cod` a) C_ C) /\ (o` a) C_ R)} e. _V
154	19, 20, 153	fvopab4 4743	1 |- (T e. Cat -> ( SubCat ` T) = {a e. Cat | ((id` a) C_ J /\ ((dom` a) C_ D /\ (cod` a) C_ C) /\ (o` a) C_ R)})

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  {crab 2108  _Vcvv 2292   C_ wss 2593  ~Pcpw 3032  <.cop 3046   X. cxp 3984  dom cdm 3986  ran crn 3987   o. ccom 3990  Rel wrel 3991  Fun wfun 3992  -->wf 3994  -onto->wfo 3996  ` cfv 3998  1stc1st 5018  2ndc2nd 5019  domcdom_ 15059  codccod_ 15060  idcid_ 15061  oco_ 15062   Cat ccat 15099   SubCat csubc 15191

This theorem is referenced by:  issubcata 15194  issubcatb 15195

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-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-rab 2112  df-v 2294  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-1st 5020  df-2nd 5021  df-doma 15064  df-coda 15065  df-ida 15066  df-cmpa 15067  df-cat 15100  df-subc 15192

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