Theorem cbicplem 14508

Description: Lemma for cbicp 14509.

Hypotheses

Ref	Expression
cbicplem.1	|- F = (f e. X_x e. {A}B |-> (f` A))
cbicplem.2	|- (x = A -> B = C)

Assertion

Ref	Expression
cbicplem	|- ((A e. D /\ C e. E) -> F:X_x e. {A}B-1-1-onto->C)

Distinct variable groups:   A,f,x   B,f   C,f,x   D,f,x   f,E   f,F

Proof of Theorem cbicplem

Step	Hyp	Ref	Expression
1		fvex 4689	. . . . . 6 |- (f` A) e. _V
2	1	a1i 8	. . . . 5 |- (f e. X_x e. {A}B -> (f` A) e. _V)
3	2	rgen 2159	. . . 4 |- A.f e. X_ x e. {A}B(f` A) e. _V
4		cbicplem.1	. . . . . 6 |- F = (f e. X_x e. {A}B |-> (f` A))
5	4	fopab2ga 14478	. . . . 5 |- (A.f e. X_ x e. {A}B(f` A) e. _V <-> F Fn X_x e. {A}B)
6	5	a1i 8	. . . 4 |- ((A e. D /\ C e. E) -> (A.f e. X_ x e. {A}B(f` A) e. _V <-> F Fn X_x e. {A}B))
7	3, 6	mpbii 210	. . 3 |- ((A e. D /\ C e. E) -> F Fn X_x e. {A}B)
8		elixp2 5408	. . . . . . . . . 10 |- (f e. X_x e. {A}B <-> (f e. _V /\ f Fn {A} /\ A.x e. {A} (f` x) e. B))
9		3simpc 874	. . . . . . . . . . 11 |- ((f e. _V /\ f Fn {A} /\ A.x e. {A} (f` x) e. B) -> (f Fn {A} /\ A.x e. {A} (f` x) e. B))
10		visset 2295	. . . . . . . . . . . . 13 |- f e. _V
11	10	a1i 8	. . . . . . . . . . . 12 |- ((f Fn {A} /\ A.x e. {A} (f` x) e. B) -> f e. _V)
12		simpl 346	. . . . . . . . . . . 12 |- ((f Fn {A} /\ A.x e. {A} (f` x) e. B) -> f Fn {A})
13		simpr 350	. . . . . . . . . . . 12 |- ((f Fn {A} /\ A.x e. {A} (f` x) e. B) -> A.x e. {A} (f` x) e. B)
14	11, 12, 13	3jca 1050	. . . . . . . . . . 11 |- ((f Fn {A} /\ A.x e. {A} (f` x) e. B) -> (f e. _V /\ f Fn {A} /\ A.x e. {A} (f` x) e. B))
15	9, 14	impbii 174	. . . . . . . . . 10 |- ((f e. _V /\ f Fn {A} /\ A.x e. {A} (f` x) e. B) <-> (f Fn {A} /\ A.x e. {A} (f` x) e. B))
16	8, 15	bitri 190	. . . . . . . . 9 |- (f e. X_x e. {A}B <-> (f Fn {A} /\ A.x e. {A} (f` x) e. B))
17	16	anbi1i 539	. . . . . . . 8 |- ((f e. X_x e. {A}B /\ y = (f` A)) <-> ((f Fn {A} /\ A.x e. {A} (f` x) e. B) /\ y = (f` A)))
18	17	exbii 1398	. . . . . . 7 |- (E.f(f e. X_x e. {A}B /\ y = (f` A)) <-> E.f((f Fn {A} /\ A.x e. {A} (f` x) e. B) /\ y = (f` A)))
19	18	a1i 8	. . . . . 6 |- ((A e. D /\ C e. E) -> (E.f(f e. X_x e. {A}B /\ y = (f` A)) <-> E.f((f Fn {A} /\ A.x e. {A} (f` x) e. B) /\ y = (f` A))))
20		df-rex 2110	. . . . . 6 |- (E.f e. X_ x e. {A}By = (f` A) <-> E.f(f e. X_x e. {A}B /\ y = (f` A)))
21	19, 20	syl5bb 591	. . . . 5 |- ((A e. D /\ C e. E) -> (E.f e. X_ x e. {A}By = (f` A) <-> E.f((f Fn {A} /\ A.x e. {A} (f` x) e. B) /\ y = (f` A))))
22	21	abbidv 2008	. . . 4 |- ((A e. D /\ C e. E) -> {y | E.f e. X_ x e. {A}By = (f` A)} = {y | E.f((f Fn {A} /\ A.x e. {A} (f` x) e. B) /\ y = (f` A))})
23	4	rnmpt 10163	. . . . 5 |- ran F = {y | E.f e. X_ x e. {A}By = (f` A)}
24	23	a1i 8	. . . 4 |- ((A e. D /\ C e. E) -> ran F = {y | E.f e. X_ x e. {A}By = (f` A)})
25		visset 2295	. . . . . . . . . . . . . . 15 |- y e. _V
26	25	a1i 8	. . . . . . . . . . . . . 14 |- (x e. {A} -> y e. _V)
27	26	rgen 2159	. . . . . . . . . . . . 13 |- A.x e. {A}y e. _V
28		eqid 1884	. . . . . . . . . . . . . 14 |- (x e. {A} |-> y) = (x e. {A} |-> y)
29	28	mptfn 10162	. . . . . . . . . . . . 13 |- (A.x e. {A}y e. _V <-> (x e. {A} |-> y) Fn {A})
30	27, 29	mpbi 206	. . . . . . . . . . . 12 |- (x e. {A} |-> y) Fn {A}
31	30	a1i 8	. . . . . . . . . . 11 |- (((A e. D /\ C e. E) /\ y e. C) -> (x e. {A} |-> y) Fn {A})
32		eqidd 1885	. . . . . . . . . . . . . 14 |- (x = A -> y = y)
33	32, 28	fvmptg 5014	. . . . . . . . . . . . 13 |- ((A e. {A} /\ y e. C) -> ((x e. {A} |-> y)` A) = y)
34		eleq1a 1966	. . . . . . . . . . . . . 14 |- (y e. C -> (((x e. {A} |-> y)` A) = y -> ((x e. {A} |-> y)` A) e. C))
35	34	adantl 424	. . . . . . . . . . . . 13 |- ((A e. {A} /\ y e. C) -> (((x e. {A} |-> y)` A) = y -> ((x e. {A} |-> y)` A) e. C))
36	33, 35	mpd 29	. . . . . . . . . . . 12 |- ((A e. {A} /\ y e. C) -> ((x e. {A} |-> y)` A) e. C)
37		snidg 3067	. . . . . . . . . . . . 13 |- (A e. D -> A e. {A})
38	37	adantr 425	. . . . . . . . . . . 12 |- ((A e. D /\ C e. E) -> A e. {A})
39	36, 38	sylan 497	. . . . . . . . . . 11 |- (((A e. D /\ C e. E) /\ y e. C) -> ((x e. {A} |-> y)` A) e. C)
40	33	eqcomd 1889	. . . . . . . . . . . 12 |- ((A e. {A} /\ y e. C) -> y = ((x e. {A} |-> y)` A))
41	40, 38	sylan 497	. . . . . . . . . . 11 |- (((A e. D /\ C e. E) /\ y e. C) -> y = ((x e. {A} |-> y)` A))
42	31, 39, 41	jca31 311	. . . . . . . . . 10 |- (((A e. D /\ C e. E) /\ y e. C) -> (((x e. {A} |-> y) Fn {A} /\ ((x e. {A} |-> y)` A) e. C) /\ y = ((x e. {A} |-> y)` A)))
43	42	ex 402	. . . . . . . . 9 |- ((A e. D /\ C e. E) -> (y e. C -> (((x e. {A} |-> y) Fn {A} /\ ((x e. {A} |-> y)` A) e. C) /\ y = ((x e. {A} |-> y)` A))))
44		snex 3492	. . . . . . . . . . 11 |- {A} e. _V
45		mptexg 5012	. . . . . . . . . . 11 |- ({A} e. _V -> (x e. {A} |-> y) e. _V)
46	44, 45	ax-mp 7	. . . . . . . . . 10 |- (x e. {A} |-> y) e. _V
47		fneq1 4503	. . . . . . . . . . . 12 |- (f = (x e. {A} |-> y) -> (f Fn {A} <-> (x e. {A} |-> y) Fn {A}))
48		fveq1 4680	. . . . . . . . . . . . 13 |- (f = (x e. {A} |-> y) -> (f` A) = ((x e. {A} |-> y)` A))
49	48	eleq1d 1963	. . . . . . . . . . . 12 |- (f = (x e. {A} |-> y) -> ((f` A) e. C <-> ((x e. {A} |-> y)` A) e. C))
50	47, 49	anbi12d 690	. . . . . . . . . . 11 |- (f = (x e. {A} |-> y) -> ((f Fn {A} /\ (f` A) e. C) <-> ((x e. {A} |-> y) Fn {A} /\ ((x e. {A} |-> y)` A) e. C)))
51	48	eqeq2d 1895	. . . . . . . . . . 11 |- (f = (x e. {A} |-> y) -> (y = (f` A) <-> y = ((x e. {A} |-> y)` A)))
52	50, 51	anbi12d 690	. . . . . . . . . 10 |- (f = (x e. {A} |-> y) -> (((f Fn {A} /\ (f` A) e. C) /\ y = (f` A)) <-> (((x e. {A} |-> y) Fn {A} /\ ((x e. {A} |-> y)` A) e. C) /\ y = ((x e. {A} |-> y)` A))))
53	46, 52	cla4ev 2371	. . . . . . . . 9 |- ((((x e. {A} |-> y) Fn {A} /\ ((x e. {A} |-> y)` A) e. C) /\ y = ((x e. {A} |-> y)` A)) -> E.f((f Fn {A} /\ (f` A) e. C) /\ y = (f` A)))
54	43, 53	syl6 25	. . . . . . . 8 |- ((A e. D /\ C e. E) -> (y e. C -> E.f((f Fn {A} /\ (f` A) e. C) /\ y = (f` A))))
55		eleq1a 1966	. . . . . . . . . . 11 |- ((f` A) e. C -> (y = (f` A) -> y e. C))
56	55	adantl 424	. . . . . . . . . 10 |- ((f Fn {A} /\ (f` A) e. C) -> (y = (f` A) -> y e. C))
57	56	imp 377	. . . . . . . . 9 |- (((f Fn {A} /\ (f` A) e. C) /\ y = (f` A)) -> y e. C)
58	57	19.23aiv 1674	. . . . . . . 8 |- (E.f((f Fn {A} /\ (f` A) e. C) /\ y = (f` A)) -> y e. C)
59	54, 58	impbid1 575	. . . . . . 7 |- ((A e. D /\ C e. E) -> (y e. C <-> E.f((f Fn {A} /\ (f` A) e. C) /\ y = (f` A))))
60		ralsng 3085	. . . . . . . . . . . 12 |- (A e. D -> (A.x e. {A} (f` x) e. B <-> [A / x](f` x) e. B))
61		ax-17 1317	. . . . . . . . . . . . . 14 |- ((f` A) e. C -> A.x(f` A) e. C)
62	61	a1i 8	. . . . . . . . . . . . 13 |- (A e. D -> ((f` A) e. C -> A.x(f` A) e. C))
63		fveq2 4681	. . . . . . . . . . . . . 14 |- (x = A -> (f` x) = (f` A))
64		cbicplem.2	. . . . . . . . . . . . . 14 |- (x = A -> B = C)
65	63, 64	eleq12d 1965	. . . . . . . . . . . . 13 |- (x = A -> ((f` x) e. B <-> (f` A) e. C))
66	62, 65	sbciegf 2483	. . . . . . . . . . . 12 |- (A e. D -> ([A / x](f` x) e. B <-> (f` A) e. C))
67	60, 66	bitr2d 588	. . . . . . . . . . 11 |- (A e. D -> ((f` A) e. C <-> A.x e. {A} (f` x) e. B))
68	67	adantr 425	. . . . . . . . . 10 |- ((A e. D /\ C e. E) -> ((f` A) e. C <-> A.x e. {A} (f` x) e. B))
69	68	anbi2d 678	. . . . . . . . 9 |- ((A e. D /\ C e. E) -> ((f Fn {A} /\ (f` A) e. C) <-> (f Fn {A} /\ A.x e. {A} (f` x) e. B)))
70	69	anbi1d 679	. . . . . . . 8 |- ((A e. D /\ C e. E) -> (((f Fn {A} /\ (f` A) e. C) /\ y = (f` A)) <-> ((f Fn {A} /\ A.x e. {A} (f` x) e. B) /\ y = (f` A))))
71	70	exbidv 1657	. . . . . . 7 |- ((A e. D /\ C e. E) -> (E.f((f Fn {A} /\ (f` A) e. C) /\ y = (f` A)) <-> E.f((f Fn {A} /\ A.x e. {A} (f` x) e. B) /\ y = (f` A))))
72	59, 71	bitrd 587	. . . . . 6 |- ((A e. D /\ C e. E) -> (y e. C <-> E.f((f Fn {A} /\ A.x e. {A} (f` x) e. B) /\ y = (f` A))))
73	72	19.21aiv 1664	. . . . 5 |- ((A e. D /\ C e. E) -> A.y(y e. C <-> E.f((f Fn {A} /\ A.x e. {A} (f` x) e. B) /\ y = (f` A))))
74		abeq2 1999	. . . . 5 |- (C = {y | E.f((f Fn {A} /\ A.x e. {A} (f` x) e. B) /\ y = (f` A))} <-> A.y(y e. C <-> E.f((f Fn {A} /\ A.x e. {A} (f` x) e. B) /\ y = (f` A))))
75	73, 74	sylibr 217	. . . 4 |- ((A e. D /\ C e. E) -> C = {y | E.f((f Fn {A} /\ A.x e. {A} (f` x) e. B) /\ y = (f` A))})
76	22, 24, 75	3eqtr4d 1937	. . 3 |- ((A e. D /\ C e. E) -> ran F = C)
77		fvex 4689	. . . . . . 7 |- (g` A) e. _V
78	4	fvopab2b 14476	. . . . . . . . . 10 |- ((f e. X_x e. {A}B /\ (f` A) e. _V) -> (F` f) = (f` A))
79	78	ex 402	. . . . . . . . 9 |- (f e. X_x e. {A}B -> ((f` A) e. _V -> (F` f) = (f` A)))
80		fveq1 4680	. . . . . . . . . . . 12 |- (f = g -> (f` A) = (g` A))
81	4, 80	cmpbvb 14477	. . . . . . . . . . 11 |- F = (g e. X_x e. {A}B |-> (g` A))
82	81	fvopab2b 14476	. . . . . . . . . 10 |- ((g e. X_x e. {A}B /\ (g` A) e. _V) -> (F` g) = (g` A))
83	82	ex 402	. . . . . . . . 9 |- (g e. X_x e. {A}B -> ((g` A) e. _V -> (F` g) = (g` A)))
84	79, 83	im2anan9 622	. . . . . . . 8 |- ((f e. X_x e. {A}B /\ g e. X_x e. {A}B) -> (((f` A) e. _V /\ (g` A) e. _V) -> ((F` f) = (f` A) /\ (F` g) = (g` A))))
85		eqeq1 1890	. . . . . . . . . . . . 13 |- ((F` f) = (f` A) -> ((F` f) = (F` g) <-> (f` A) = (F` g)))
86	85	imbi1d 675	. . . . . . . . . . . 12 |- ((F` f) = (f` A) -> (((F` f) = (F` g) -> f = g) <-> ((f` A) = (F` g) -> f = g)))
87	86	imbi2d 674	. . . . . . . . . . 11 |- ((F` f) = (f` A) -> (((A e. D /\ C e. E) -> ((F` f) = (F` g) -> f = g)) <-> ((A e. D /\ C e. E) -> ((f` A) = (F` g) -> f = g))))
88	87	imbi2d 674	. . . . . . . . . 10 |- ((F` f) = (f` A) -> (((f e. X_x e. {A}B /\ g e. X_x e. {A}B) -> ((A e. D /\ C e. E) -> ((F` f) = (F` g) -> f = g))) <-> ((f e. X_x e. {A}B /\ g e. X_x e. {A}B) -> ((A e. D /\ C e. E) -> ((f` A) = (F` g) -> f = g)))))
89		eqeq2 1893	. . . . . . . . . . . . 13 |- ((F` g) = (g` A) -> ((f` A) = (F` g) <-> (f` A) = (g` A)))
90	89	imbi1d 675	. . . . . . . . . . . 12 |- ((F` g) = (g` A) -> (((f` A) = (F` g) -> f = g) <-> ((f` A) = (g` A) -> f = g)))
91	90	imbi2d 674	. . . . . . . . . . 11 |- ((F` g) = (g` A) -> (((A e. D /\ C e. E) -> ((f` A) = (F` g) -> f = g)) <-> ((A e. D /\ C e. E) -> ((f` A) = (g` A) -> f = g))))
92		eqidd 1885	. . . . . . . . . . . . . 14 |- (((f e. X_x e. {A}B /\ g e. X_x e. {A}B) /\ (A e. D /\ C e. E) /\ (f` A) = (g` A)) -> {A} = {A})
93		elsn 3058	. . . . . . . . . . . . . . . . . 18 |- (x e. {A} <-> x = A)
94		fveq2 4681	. . . . . . . . . . . . . . . . . . . 20 |- (x = A -> (g` x) = (g` A))
95	63, 94	eqeq12d 1899	. . . . . . . . . . . . . . . . . . 19 |- (x = A -> ((f` x) = (g` x) <-> (f` A) = (g` A)))
96	95	biimprd 171	. . . . . . . . . . . . . . . . . 18 |- (x = A -> ((f` A) = (g` A) -> (f` x) = (g` x)))
97	93, 96	sylbi 216	. . . . . . . . . . . . . . . . 17 |- (x e. {A} -> ((f` A) = (g` A) -> (f` x) = (g` x)))
98	97	com12 14	. . . . . . . . . . . . . . . 16 |- ((f` A) = (g` A) -> (x e. {A} -> (f` x) = (g` x)))
99	98	r19.21aiv 2175	. . . . . . . . . . . . . . 15 |- ((f` A) = (g` A) -> A.x e. {A} (f` x) = (g` x))
100	99	3ad2ant3 899	. . . . . . . . . . . . . 14 |- (((f e. X_x e. {A}B /\ g e. X_x e. {A}B) /\ (A e. D /\ C e. E) /\ (f` A) = (g` A)) -> A.x e. {A} (f` x) = (g` x))
101	92, 100	jca 310	. . . . . . . . . . . . 13 |- (((f e. X_x e. {A}B /\ g e. X_x e. {A}B) /\ (A e. D /\ C e. E) /\ (f` A) = (g` A)) -> ({A} = {A} /\ A.x e. {A} (f` x) = (g` x)))
102		elixp2a 14493	. . . . . . . . . . . . . . . 16 |- (f e. X_x e. {A}B -> f Fn {A})
103		elixp2a 14493	. . . . . . . . . . . . . . . 16 |- (g e. X_x e. {A}B -> g Fn {A})
104	102, 103	anim12i 360	. . . . . . . . . . . . . . 15 |- ((f e. X_x e. {A}B /\ g e. X_x e. {A}B) -> (f Fn {A} /\ g Fn {A}))
105	104	3ad2ant1 897	. . . . . . . . . . . . . 14 |- (((f e. X_x e. {A}B /\ g e. X_x e. {A}B) /\ (A e. D /\ C e. E) /\ (f` A) = (g` A)) -> (f Fn {A} /\ g Fn {A}))
106		eqfnfv 4766	. . . . . . . . . . . . . 14 |- ((f Fn {A} /\ g Fn {A}) -> (f = g <-> ({A} = {A} /\ A.x e. {A} (f` x) = (g` x))))
107	105, 106	syl 12	. . . . . . . . . . . . 13 |- (((f e. X_x e. {A}B /\ g e. X_x e. {A}B) /\ (A e. D /\ C e. E) /\ (f` A) = (g` A)) -> (f = g <-> ({A} = {A} /\ A.x e. {A} (f` x) = (g` x))))
108	101, 107	mpbird 213	. . . . . . . . . . . 12 |- (((f e. X_x e. {A}B /\ g e. X_x e. {A}B) /\ (A e. D /\ C e. E) /\ (f` A) = (g` A)) -> f = g)
109	108	3exp 1066	. . . . . . . . . . 11 |- ((f e. X_x e. {A}B /\ g e. X_x e. {A}B) -> ((A e. D /\ C e. E) -> ((f` A) = (g` A) -> f = g)))
110	91, 109	syl5bir 227	. . . . . . . . . 10 |- ((F` g) = (g` A) -> ((f e. X_x e. {A}B /\ g e. X_x e. {A}B) -> ((A e. D /\ C e. E) -> ((f` A) = (F` g) -> f = g))))
111	88, 110	syl5bir 227	. . . . . . . . 9 |- ((F` f) = (f` A) -> ((F` g) = (g` A) -> ((f e. X_x e. {A}B /\ g e. X_x e. {A}B) -> ((A e. D /\ C e. E) -> ((F` f) = (F` g) -> f = g)))))
112	111	imp 377	. . . . . . . 8 |- (((F` f) = (f` A) /\ (F` g) = (g` A)) -> ((f e. X_x e. {A}B /\ g e. X_x e. {A}B) -> ((A e. D /\ C e. E) -> ((F` f) = (F` g) -> f = g))))
113	84, 112	syl6com 64	. . . . . . 7 |- (((f` A) e. _V /\ (g` A) e. _V) -> ((f e. X_x e. {A}B /\ g e. X_x e. {A}B) -> ((f e. X_x e. {A}B /\ g e. X_x e. {A}B) -> ((A e. D /\ C e. E) -> ((F` f) = (F` g) -> f = g)))))
114	1, 77, 113	mp2an 761	. . . . . 6 |- ((f e. X_x e. {A}B /\ g e. X_x e. {A}B) -> ((f e. X_x e. {A}B /\ g e. X_x e. {A}B) -> ((A e. D /\ C e. E) -> ((F` f) = (F` g) -> f = g))))
115	114	pm2.43i 78	. . . . 5 |- ((f e. X_x e. {A}B /\ g e. X_x e. {A}B) -> ((A e. D /\ C e. E) -> ((F` f) = (F` g) -> f = g)))
116	115	com12 14	. . . 4 |- ((A e. D /\ C e. E) -> ((f e. X_x e. {A}B /\ g e. X_x e. {A}B) -> ((F` f) = (F` g) -> f = g)))
117	116	r19.21aivv 2183	. . 3 |- ((A e. D /\ C e. E) -> A.f e. X_ x e. {A}BA.g e. X_ x e. {A}B((F` f) = (F` g) -> f = g))
118	7, 76, 117	3jca 1050	. 2 |- ((A e. D /\ C e. E) -> (F Fn X_x e. {A}B /\ ran F = C /\ A.f e. X_ x e. {A}BA.g e. X_ x e. {A}B((F` f) = (F` g) -> f = g)))
119		dff1o6 4853	. 2 |- (F:X_x e. {A}B-1-1-onto->C <-> (F Fn X_x e. {A}B /\ ran F = C /\ A.f e. X_ x e. {A}BA.g e. X_ x e. {A}B((F` f) = (F` g) -> f = g)))
120	118, 119	sylibr 217	1 |- ((A e. D /\ C e. E) -> F:X_x e. {A}B-1-1-onto->C)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  [wsbc 1534  {cab 1871  A.wral 2105  E.wrex 2106  _Vcvv 2292  {csn 3044  ran crn 3987   Fn wfn 3993  -1-1-onto->wf1o 3997  ` cfv 3998   e. cmpt 5004  X_cixp 5406

This theorem is referenced by:  cbicp 14509

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-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-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-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-mpt 5006  df-ixp 5407

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