Theorem subtop 8916

Description: A subspace topology is a topology. Definition of subspace topology in [Munkres] p. 89. Y is normally a subset of the base set of J. (Contributed by FL, 15-Apr-2007.)

Assertion

Ref	Expression
subtop	|- (J e. Top -> {x | E.y e. J x = (y i^i Y)} e. Top)

Distinct variable groups:   x,y,J   x,Y,y

Proof of Theorem subtop

Step	Hyp	Ref	Expression
1		abrexexg 4837	. . 3 |- (J e. Top -> {x | E.y e. J x = (y i^i Y)} e. _V)
2		istopg 8865	. . 3 |- ({x | E.y e. J x = (y i^i Y)} e. _V -> ({x | E.y e. J x = (y i^i Y)} e. Top <-> (A.a(a C_ {x | E.y e. J x = (y i^i Y)} -> U.a e. {x | E.y e. J x = (y i^i Y)}) /\ A.a e. {x | E.y e. J x = (y i^i Y)}A.b e. {x | E.y e. J x = (y i^i Y)} (a i^i b) e. {x | E.y e. J x = (y i^i Y)})))
3	1, 2	syl 12	. 2 |- (J e. Top -> ({x | E.y e. J x = (y i^i Y)} e. Top <-> (A.a(a C_ {x | E.y e. J x = (y i^i Y)} -> U.a e. {x | E.y e. J x = (y i^i Y)}) /\ A.a e. {x | E.y e. J x = (y i^i Y)}A.b e. {x | E.y e. J x = (y i^i Y)} (a i^i b) e. {x | E.y e. J x = (y i^i Y)})))
4		funopabeq 4456	. . . . . . . . . . . 12 |- Fun {<.y, x>. | x = (y i^i Y)}
5		ssimaexg 4730	. . . . . . . . . . . 12 |- ((J e. Top /\ Fun {<.y, x>. | x = (y i^i Y)} /\ a C_ ({<.y, x>. | x = (y i^i Y)}"J)) -> E.z(z C_ J /\ a = ({<.y, x>. | x = (y i^i Y)}"z)))
6	4, 5	mp3an2 1179	. . . . . . . . . . 11 |- ((J e. Top /\ a C_ ({<.y, x>. | x = (y i^i Y)}"J)) -> E.z(z C_ J /\ a = ({<.y, x>. | x = (y i^i Y)}"z)))
7	6	ex 402	. . . . . . . . . 10 |- (J e. Top -> (a C_ ({<.y, x>. | x = (y i^i Y)}"J) -> E.z(z C_ J /\ a = ({<.y, x>. | x = (y i^i Y)}"z))))
8		df-ima 4007	. . . . . . . . . . . 12 |- ({<.y, x>. | x = (y i^i Y)}"J) = ran ({<.y, x>. | x = (y i^i Y)} |` J)
9		resopab 4252	. . . . . . . . . . . . 13 |- ({<.y, x>. | x = (y i^i Y)} |` J) = {<.y, x>. | (y e. J /\ x = (y i^i Y))}
10	9	rneqi 4187	. . . . . . . . . . . 12 |- ran ({<.y, x>. | x = (y i^i Y)} |` J) = ran {<.y, x>. | (y e. J /\ x = (y i^i Y))}
11	8, 10	eqtri 1908	. . . . . . . . . . 11 |- ({<.y, x>. | x = (y i^i Y)}"J) = ran {<.y, x>. | (y e. J /\ x = (y i^i Y))}
12	11	sseq2i 2642	. . . . . . . . . 10 |- (a C_ ({<.y, x>. | x = (y i^i Y)}"J) <-> a C_ ran {<.y, x>. | (y e. J /\ x = (y i^i Y))})
13		df-ima 4007	. . . . . . . . . . . . . 14 |- ({<.y, x>. | x = (y i^i Y)}"z) = ran ({<.y, x>. | x = (y i^i Y)} |` z)
14		resopab 4252	. . . . . . . . . . . . . . 15 |- ({<.y, x>. | x = (y i^i Y)} |` z) = {<.y, x>. | (y e. z /\ x = (y i^i Y))}
15	14	rneqi 4187	. . . . . . . . . . . . . 14 |- ran ({<.y, x>. | x = (y i^i Y)} |` z) = ran {<.y, x>. | (y e. z /\ x = (y i^i Y))}
16		rnopab2 4202	. . . . . . . . . . . . . 14 |- ran {<.y, x>. | (y e. z /\ x = (y i^i Y))} = {x | E.y e. z x = (y i^i Y)}
17	13, 15, 16	3eqtri 1912	. . . . . . . . . . . . 13 |- ({<.y, x>. | x = (y i^i Y)}"z) = {x | E.y e. z x = (y i^i Y)}
18	17	eqeq2i 1894	. . . . . . . . . . . 12 |- (a = ({<.y, x>. | x = (y i^i Y)}"z) <-> a = {x | E.y e. z x = (y i^i Y)})
19	18	anbi2i 538	. . . . . . . . . . 11 |- ((z C_ J /\ a = ({<.y, x>. | x = (y i^i Y)}"z)) <-> (z C_ J /\ a = {x | E.y e. z x = (y i^i Y)}))
20	19	exbii 1398	. . . . . . . . . 10 |- (E.z(z C_ J /\ a = ({<.y, x>. | x = (y i^i Y)}"z)) <-> E.z(z C_ J /\ a = {x | E.y e. z x = (y i^i Y)}))
21	7, 12, 20	3imtr3g 611	. . . . . . . . 9 |- (J e. Top -> (a C_ ran {<.y, x>. | (y e. J /\ x = (y i^i Y))} -> E.z(z C_ J /\ a = {x | E.y e. z x = (y i^i Y)})))
22		rnopab2 4202	. . . . . . . . . . 11 |- ran {<.y, x>. | (y e. J /\ x = (y i^i Y))} = {x | E.y e. J x = (y i^i Y)}
23	22	eqcomi 1888	. . . . . . . . . 10 |- {x | E.y e. J x = (y i^i Y)} = ran {<.y, x>. | (y e. J /\ x = (y i^i Y))}
24	23	sseq2i 2642	. . . . . . . . 9 |- (a C_ {x | E.y e. J x = (y i^i Y)} <-> a C_ ran {<.y, x>. | (y e. J /\ x = (y i^i Y))})
25	21, 24	syl5ib 223	. . . . . . . 8 |- (J e. Top -> (a C_ {x | E.y e. J x = (y i^i Y)} -> E.z(z C_ J /\ a = {x | E.y e. z x = (y i^i Y)})))
26	25	imp 377	. . . . . . 7 |- ((J e. Top /\ a C_ {x | E.y e. J x = (y i^i Y)}) -> E.z(z C_ J /\ a = {x | E.y e. z x = (y i^i Y)}))
27		unieq 3185	. . . . . . . . . 10 |- (a = {x | E.y e. z x = (y i^i Y)} -> U.a = U.{x | E.y e. z x = (y i^i Y)})
28		dfiun2g 3283	. . . . . . . . . . . . 13 |- (A.y e. z (y i^i Y) e. _V -> U_y e. z (y i^i Y) = U.{x | E.y e. z x = (y i^i Y)})
29	28	eqcomd 1889	. . . . . . . . . . . 12 |- (A.y e. z (y i^i Y) e. _V -> U.{x | E.y e. z x = (y i^i Y)} = U_y e. z (y i^i Y))
30		inex1g 3454	. . . . . . . . . . . 12 |- (y e. z -> (y i^i Y) e. _V)
31	29, 30	mprg 2162	. . . . . . . . . . 11 |- U.{x | E.y e. z x = (y i^i Y)} = U_y e. z (y i^i Y)
32		incom 2787	. . . . . . . . . . . . 13 |- (y i^i Y) = (Y i^i y)
33	32	a1i 8	. . . . . . . . . . . 12 |- (y e. z -> (y i^i Y) = (Y i^i y))
34	33	iuneq2i 3276	. . . . . . . . . . 11 |- U_y e. z (y i^i Y) = U_y e. z (Y i^i y)
35		iunin2 3320	. . . . . . . . . . . 12 |- U_y e. z (Y i^i y) = (Y i^i U_y e. z y)
36		uniiun 3306	. . . . . . . . . . . . 13 |- U.z = U_y e. z y
37	36	ineq2i 2793	. . . . . . . . . . . 12 |- (Y i^i U.z) = (Y i^i U_y e. z y)
38		incom 2787	. . . . . . . . . . . 12 |- (Y i^i U.z) = (U.z i^i Y)
39	35, 37, 38	3eqtr2i 1915	. . . . . . . . . . 11 |- U_y e. z (Y i^i y) = (U.z i^i Y)
40	31, 34, 39	3eqtri 1912	. . . . . . . . . 10 |- U.{x | E.y e. z x = (y i^i Y)} = (U.z i^i Y)
41	27, 40	syl6eq 1944	. . . . . . . . 9 |- (a = {x | E.y e. z x = (y i^i Y)} -> U.a = (U.z i^i Y))
42	41	anim2i 362	. . . . . . . 8 |- ((z C_ J /\ a = {x | E.y e. z x = (y i^i Y)}) -> (z C_ J /\ U.a = (U.z i^i Y)))
43	42	eximi 1387	. . . . . . 7 |- (E.z(z C_ J /\ a = {x | E.y e. z x = (y i^i Y)}) -> E.z(z C_ J /\ U.a = (U.z i^i Y)))
44	26, 43	syl 12	. . . . . 6 |- ((J e. Top /\ a C_ {x | E.y e. J x = (y i^i Y)}) -> E.z(z C_ J /\ U.a = (U.z i^i Y)))
45		snex 3492	. . . . . . . . . . 11 |- {y} e. _V
46		sseq1 2637	. . . . . . . . . . . 12 |- (z = {y} -> (z C_ J <-> {y} C_ J))
47		unieq 3185	. . . . . . . . . . . . . . 15 |- (z = {y} -> U.z = U.{y})
48		visset 2295	. . . . . . . . . . . . . . . 16 |- y e. _V
49	48	unisn 3193	. . . . . . . . . . . . . . 15 |- U.{y} = y
50	47, 49	syl6eq 1944	. . . . . . . . . . . . . 14 |- (z = {y} -> U.z = y)
51	50	ineq1d 2795	. . . . . . . . . . . . 13 |- (z = {y} -> (U.z i^i Y) = (y i^i Y))
52	51	eqeq2d 1895	. . . . . . . . . . . 12 |- (z = {y} -> (U.a = (U.z i^i Y) <-> U.a = (y i^i Y)))
53	46, 52	anbi12d 690	. . . . . . . . . . 11 |- (z = {y} -> ((z C_ J /\ U.a = (U.z i^i Y)) <-> ({y} C_ J /\ U.a = (y i^i Y))))
54	45, 53	cla4ev 2371	. . . . . . . . . 10 |- (({y} C_ J /\ U.a = (y i^i Y)) -> E.z(z C_ J /\ U.a = (U.z i^i Y)))
55		snssi 3129	. . . . . . . . . 10 |- (y e. J -> {y} C_ J)
56	54, 55	sylan 497	. . . . . . . . 9 |- ((y e. J /\ U.a = (y i^i Y)) -> E.z(z C_ J /\ U.a = (U.z i^i Y)))
57	56	r19.23aiva 2212	. . . . . . . 8 |- (E.y e. J U.a = (y i^i Y) -> E.z(z C_ J /\ U.a = (U.z i^i Y)))
58		uniopn 8867	. . . . . . . . . . 11 |- ((J e. Top /\ z C_ J) -> U.z e. J)
59		ineq1 2789	. . . . . . . . . . . . . 14 |- (y = U.z -> (y i^i Y) = (U.z i^i Y))
60	59	eqeq2d 1895	. . . . . . . . . . . . 13 |- (y = U.z -> (U.a = (y i^i Y) <-> U.a = (U.z i^i Y)))
61	60	rcla4ev 2381	. . . . . . . . . . . 12 |- ((U.z e. J /\ U.a = (U.z i^i Y)) -> E.y e. J U.a = (y i^i Y))
62	61	ex 402	. . . . . . . . . . 11 |- (U.z e. J -> (U.a = (U.z i^i Y) -> E.y e. J U.a = (y i^i Y)))
63	58, 62	syl 12	. . . . . . . . . 10 |- ((J e. Top /\ z C_ J) -> (U.a = (U.z i^i Y) -> E.y e. J U.a = (y i^i Y)))
64	63	expimpd 404	. . . . . . . . 9 |- (J e. Top -> ((z C_ J /\ U.a = (U.z i^i Y)) -> E.y e. J U.a = (y i^i Y)))
65	64	19.23adv 1584	. . . . . . . 8 |- (J e. Top -> (E.z(z C_ J /\ U.a = (U.z i^i Y)) -> E.y e. J U.a = (y i^i Y)))
66	57, 65	impbid2 576	. . . . . . 7 |- (J e. Top -> (E.y e. J U.a = (y i^i Y) <-> E.z(z C_ J /\ U.a = (U.z i^i Y))))
67	66	adantr 425	. . . . . 6 |- ((J e. Top /\ a C_ {x | E.y e. J x = (y i^i Y)}) -> (E.y e. J U.a = (y i^i Y) <-> E.z(z C_ J /\ U.a = (U.z i^i Y))))
68	44, 67	mpbird 213	. . . . 5 |- ((J e. Top /\ a C_ {x | E.y e. J x = (y i^i Y)}) -> E.y e. J U.a = (y i^i Y))
69		visset 2295	. . . . . . 7 |- a e. _V
70	69	uniex 3794	. . . . . 6 |- U.a e. _V
71		eqeq1 1890	. . . . . . 7 |- (x = U.a -> (x = (y i^i Y) <-> U.a = (y i^i Y)))
72	71	rexbidv 2124	. . . . . 6 |- (x = U.a -> (E.y e. J x = (y i^i Y) <-> E.y e. J U.a = (y i^i Y)))
73	70, 72	elab 2403	. . . . 5 |- (U.a e. {x | E.y e. J x = (y i^i Y)} <-> E.y e. J U.a = (y i^i Y))
74	68, 73	sylibr 217	. . . 4 |- ((J e. Top /\ a C_ {x | E.y e. J x = (y i^i Y)}) -> U.a e. {x | E.y e. J x = (y i^i Y)})
75	74	ex 402	. . 3 |- (J e. Top -> (a C_ {x | E.y e. J x = (y i^i Y)} -> U.a e. {x | E.y e. J x = (y i^i Y)}))
76	75	19.21aiv 1664	. 2 |- (J e. Top -> A.a(a C_ {x | E.y e. J x = (y i^i Y)} -> U.a e. {x | E.y e. J x = (y i^i Y)}))
77		ineq1 2789	. . . . . . . . . . . . . . . . . . 19 |- (y = (c i^i d) -> (y i^i Y) = ((c i^i d) i^i Y))
78	77	eqeq2d 1895	. . . . . . . . . . . . . . . . . 18 |- (y = (c i^i d) -> ((a i^i b) = (y i^i Y) <-> (a i^i b) = ((c i^i d) i^i Y)))
79	78	rcla4ev 2381	. . . . . . . . . . . . . . . . 17 |- (((c i^i d) e. J /\ (a i^i b) = ((c i^i d) i^i Y)) -> E.y e. J (a i^i b) = (y i^i Y))
80		inopn 8869	. . . . . . . . . . . . . . . . . 18 |- ((J e. Top /\ c e. J /\ d e. J) -> (c i^i d) e. J)
81	80	3expa 1067	. . . . . . . . . . . . . . . . 17 |- (((J e. Top /\ c e. J) /\ d e. J) -> (c i^i d) e. J)
82		ineq12 2791	. . . . . . . . . . . . . . . . . . 19 |- ((a = (c i^i Y) /\ b = (d i^i Y)) -> (a i^i b) = ((c i^i Y) i^i (d i^i Y)))
83		inindir 2810	. . . . . . . . . . . . . . . . . . 19 |- ((c i^i d) i^i Y) = ((c i^i Y) i^i (d i^i Y))
84	82, 83	syl6eqr 1946	. . . . . . . . . . . . . . . . . 18 |- ((a = (c i^i Y) /\ b = (d i^i Y)) -> (a i^i b) = ((c i^i d) i^i Y))
85	84	ancoms 484	. . . . . . . . . . . . . . . . 17 |- ((b = (d i^i Y) /\ a = (c i^i Y)) -> (a i^i b) = ((c i^i d) i^i Y))
86	79, 81, 85	syl2an 503	. . . . . . . . . . . . . . . 16 |- ((((J e. Top /\ c e. J) /\ d e. J) /\ (b = (d i^i Y) /\ a = (c i^i Y))) -> E.y e. J (a i^i b) = (y i^i Y))
87	86	exp43 415	. . . . . . . . . . . . . . 15 |- ((J e. Top /\ c e. J) -> (d e. J -> (b = (d i^i Y) -> (a = (c i^i Y) -> E.y e. J (a i^i b) = (y i^i Y)))))
88	87	com24 41	. . . . . . . . . . . . . 14 |- ((J e. Top /\ c e. J) -> (a = (c i^i Y) -> (b = (d i^i Y) -> (d e. J -> E.y e. J (a i^i b) = (y i^i Y)))))
89	88	r19.23adva 2216	. . . . . . . . . . . . 13 |- (J e. Top -> (E.c e. J a = (c i^i Y) -> (b = (d i^i Y) -> (d e. J -> E.y e. J (a i^i b) = (y i^i Y)))))
90	89	com24 41	. . . . . . . . . . . 12 |- (J e. Top -> (d e. J -> (b = (d i^i Y) -> (E.c e. J a = (c i^i Y) -> E.y e. J (a i^i b) = (y i^i Y)))))
91	90	r19.23adv 2215	. . . . . . . . . . 11 |- (J e. Top -> (E.d e. J b = (d i^i Y) -> (E.c e. J a = (c i^i Y) -> E.y e. J (a i^i b) = (y i^i Y))))
92		ineq1 2789	. . . . . . . . . . . . 13 |- (y = d -> (y i^i Y) = (d i^i Y))
93	92	eqeq2d 1895	. . . . . . . . . . . 12 |- (y = d -> (b = (y i^i Y) <-> b = (d i^i Y)))
94	93	cbvrexv 2281	. . . . . . . . . . 11 |- (E.y e. J b = (y i^i Y) <-> E.d e. J b = (d i^i Y))
95	91, 94	syl5ib 223	. . . . . . . . . 10 |- (J e. Top -> (E.y e. J b = (y i^i Y) -> (E.c e. J a = (c i^i Y) -> E.y e. J (a i^i b) = (y i^i Y))))
96		clelab 2013	. . . . . . . . . . 11 |- (a e. {x | E.y e. J x = (y i^i Y)} <-> E.x(x = a /\ E.y e. J x = (y i^i Y)))
97		eqeq1 1890	. . . . . . . . . . . . 13 |- (x = a -> (x = (y i^i Y) <-> a = (y i^i Y)))
98	97	rexbidv 2124	. . . . . . . . . . . 12 |- (x = a -> (E.y e. J x = (y i^i Y) <-> E.y e. J a = (y i^i Y)))
99	69, 98	ceqsexv 2325	. . . . . . . . . . 11 |- (E.x(x = a /\ E.y e. J x = (y i^i Y)) <-> E.y e. J a = (y i^i Y))
100		ineq1 2789	. . . . . . . . . . . . 13 |- (y = c -> (y i^i Y) = (c i^i Y))
101	100	eqeq2d 1895	. . . . . . . . . . . 12 |- (y = c -> (a = (y i^i Y) <-> a = (c i^i Y)))
102	101	cbvrexv 2281	. . . . . . . . . . 11 |- (E.y e. J a = (y i^i Y) <-> E.c e. J a = (c i^i Y))
103	96, 99, 102	3bitri 194	. . . . . . . . . 10 |- (a e. {x | E.y e. J x = (y i^i Y)} <-> E.c e. J a = (c i^i Y))
104	95, 103	syl7ib 233	. . . . . . . . 9 |- (J e. Top -> (E.y e. J b = (y i^i Y) -> (a e. {x | E.y e. J x = (y i^i Y)} -> E.y e. J (a i^i b) = (y i^i Y))))
105	104	com23 36	. . . . . . . 8 |- (J e. Top -> (a e. {x | E.y e. J x = (y i^i Y)} -> (E.y e. J b = (y i^i Y) -> E.y e. J (a i^i b) = (y i^i Y))))
106	105	imp 377	. . . . . . 7 |- ((J e. Top /\ a e. {x | E.y e. J x = (y i^i Y)}) -> (E.y e. J b = (y i^i Y) -> E.y e. J (a i^i b) = (y i^i Y)))
107		visset 2295	. . . . . . . 8 |- b e. _V
108		eqeq1 1890	. . . . . . . . 9 |- (x = b -> (x = (y i^i Y) <-> b = (y i^i Y)))
109	108	rexbidv 2124	. . . . . . . 8 |- (x = b -> (E.y e. J x = (y i^i Y) <-> E.y e. J b = (y i^i Y)))
110	107, 109	elab 2403	. . . . . . 7 |- (b e. {x | E.y e. J x = (y i^i Y)} <-> E.y e. J b = (y i^i Y))
111	106, 110	syl5ib 223	. . . . . 6 |- ((J e. Top /\ a e. {x | E.y e. J x = (y i^i Y)}) -> (b e. {x | E.y e. J x = (y i^i Y)} -> E.y e. J (a i^i b) = (y i^i Y)))
112	111	imp 377	. . . . 5 |- (((J e. Top /\ a e. {x | E.y e. J x = (y i^i Y)}) /\ b e. {x | E.y e. J x = (y i^i Y)}) -> E.y e. J (a i^i b) = (y i^i Y))
113	69	inex1 3452	. . . . . 6 |- (a i^i b) e. _V
114		eqeq1 1890	. . . . . . 7 |- (x = (a i^i b) -> (x = (y i^i Y) <-> (a i^i b) = (y i^i Y)))
115	114	rexbidv 2124	. . . . . 6 |- (x = (a i^i b) -> (E.y e. J x = (y i^i Y) <-> E.y e. J (a i^i b) = (y i^i Y)))
116	113, 115	elab 2403	. . . . 5 |- ((a i^i b) e. {x | E.y e. J x = (y i^i Y)} <-> E.y e. J (a i^i b) = (y i^i Y))
117	112, 116	sylibr 217	. . . 4 |- (((J e. Top /\ a e. {x | E.y e. J x = (y i^i Y)}) /\ b e. {x | E.y e. J x = (y i^i Y)}) -> (a i^i b) e. {x | E.y e. J x = (y i^i Y)})
118	117	r19.21aiva 2176	. . 3 |- ((J e. Top /\ a e. {x | E.y e. J x = (y i^i Y)}) -> A.b e. {x | E.y e. J x = (y i^i Y)} (a i^i b) e. {x | E.y e. J x = (y i^i Y)})
119	118	r19.21aiva 2176	. 2 |- (J e. Top -> A.a e. {x | E.y e. J x = (y i^i Y)}A.b e. {x | E.y e. J x = (y i^i Y)} (a i^i b) e. {x | E.y e. J x = (y i^i Y)})
120	3, 76, 119	mpbir2and 802	1 |- (J e. Top -> {x | E.y e. J x = (y i^i Y)} e. Top)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  A.wral 2105  E.wrex 2106  _Vcvv 2292   i^i cin 2592   C_ wss 2593  {csn 3044  U.cuni 3177  U_ciun 3255  {copab 3395  ran crn 3987   |` cres 3988  "cima 3989  Fun wfun 3992  Topctop 8857

This theorem is referenced by:  stoiglem 10250  stoi 14998

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-rab 2112  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-iun 3257  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-fv 4014  df-top 8861

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