Theorem domrancur1c 14550

Description: The currying of a mapping F whose domain is (A X. B) is a mapping whose domain is A and the range, the class of all the functions from B to ran F.

Assertion

Ref	Expression
domrancur1c	|- (((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) -> (cur1` F):A-->{f | f:B-->ran F})

Distinct variable groups:   B,f   f,F

Proof of Theorem domrancur1c

Step	Hyp	Ref	Expression
1		fss 4571	. . . . 5 |- (((F o. `'(2nd |` ({x} X. _V))):B-->ran ( F o. `'(2nd |` ({x} X. _V))) /\ ran ( F o. `'(2nd |` ({x} X. _V))) C_ ran F) -> (F o. `'(2nd |` ({x} X. _V))):B-->ran F)
2		visset 2295	. . . . . . . . . . 11 |- x e. _V
3		2ndconst 5071	. . . . . . . . . . 11 |- (x e. _V -> (2nd |` ({x} X. _V)):({x} X. _V)-1-1-onto->_V)
4	2, 3	ax-mp 7	. . . . . . . . . 10 |- (2nd |` ({x} X. _V)):({x} X. _V)-1-1-onto->_V
5		f1ocnv 4651	. . . . . . . . . 10 |- ((2nd |` ({x} X. _V)):({x} X. _V)-1-1-onto->_V -> `'(2nd |` ({x} X. _V)):_V-1-1-onto->({x} X. _V))
6	4, 5	ax-mp 7	. . . . . . . . 9 |- `'(2nd |` ({x} X. _V)):_V-1-1-onto->({x} X. _V)
7		fnfun 4510	. . . . . . . . . . . . 13 |- (F Fn (A X. B) -> Fun F)
8		funco 4457	. . . . . . . . . . . . . 14 |- ((Fun F /\ Fun `'(2nd |` ({x} X. _V))) -> Fun (F o. `'(2nd |` ({x} X. _V))))
9	8	ex 402	. . . . . . . . . . . . 13 |- (Fun F -> (Fun `'(2nd |` ({x} X. _V)) -> Fun (F o. `'(2nd |` ({x} X. _V)))))
10	7, 9	syl 12	. . . . . . . . . . . 12 |- (F Fn (A X. B) -> (Fun `'(2nd |` ({x} X. _V)) -> Fun (F o. `'(2nd |` ({x} X. _V)))))
11	10	adantl 424	. . . . . . . . . . 11 |- ((B =/= (/) /\ F Fn (A X. B)) -> (Fun `'(2nd |` ({x} X. _V)) -> Fun (F o. `'(2nd |` ({x} X. _V)))))
12	11	ad2antlr 441	. . . . . . . . . 10 |- ((((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) /\ x e. A) -> (Fun `'(2nd |` ({x} X. _V)) -> Fun (F o. `'(2nd |` ({x} X. _V)))))
13		f1ofun 4637	. . . . . . . . . 10 |- (`'(2nd |` ({x} X. _V)):_V-1-1-onto->({x} X. _V) -> Fun `'(2nd |` ({x} X. _V)))
14	12, 13	syl5com 63	. . . . . . . . 9 |- (`'(2nd |` ({x} X. _V)):_V-1-1-onto->({x} X. _V) -> ((((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) /\ x e. A) -> Fun (F o. `'(2nd |` ({x} X. _V)))))
15	6, 14	ax-mp 7	. . . . . . . 8 |- ((((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) /\ x e. A) -> Fun (F o. `'(2nd |` ({x} X. _V))))
16		dmco 4406	. . . . . . . . . 10 |- dom ( F o. `'(2nd |` ({x} X. _V))) = (`'`'(2nd |` ({x} X. _V))"dom F)
17	16	a1i 8	. . . . . . . . 9 |- ((((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) /\ x e. A) -> dom ( F o. `'(2nd |` ({x} X. _V))) = (`'`'(2nd |` ({x} X. _V))"dom F))
18		cnvcnvres 4387	. . . . . . . . . . 11 |- `'`'(2nd |` ({x} X. _V)) = (`'`'2nd |` ({x} X. _V))
19	18	imaeq1i 4261	. . . . . . . . . 10 |- (`'`'(2nd |` ({x} X. _V))"dom F) = ((`'`'2nd |` ({x} X. _V))"dom F)
20	19	a1i 8	. . . . . . . . 9 |- ((((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) /\ x e. A) -> (`'`'(2nd |` ({x} X. _V))"dom F) = ((`'`'2nd |` ({x} X. _V))"dom F))
21		df-ima 4007	. . . . . . . . . . 11 |- ((`'`'2nd |` ({x} X. _V))"dom F) = ran ((`'`'2nd |` ({x} X. _V)) |` dom F)
22	21	a1i 8	. . . . . . . . . 10 |- ((((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) /\ x e. A) -> ((`'`'2nd |` ({x} X. _V))"dom F) = ran ((`'`'2nd |` ({x} X. _V)) |` dom F))
23		resres 4228	. . . . . . . . . . . 12 |- ((`'`'2nd |` ({x} X. _V)) |` dom F) = (`'`'2nd |` (({x} X. _V) i^i dom F))
24	23	a1i 8	. . . . . . . . . . 11 |- ((((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) /\ x e. A) -> ((`'`'2nd |` ({x} X. _V)) |` dom F) = (`'`'2nd |` (({x} X. _V) i^i dom F)))
25	24	rneqd 4188	. . . . . . . . . 10 |- ((((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) /\ x e. A) -> ran ((`'`'2nd |` ({x} X. _V)) |` dom F) = ran (`'`'2nd |` (({x} X. _V) i^i dom F)))
26		rescnvcnv 4385	. . . . . . . . . . . . 13 |- (`'`'2nd |` (({x} X. _V) i^i dom F)) = (2nd |` (({x} X. _V) i^i dom F))
27	26	a1i 8	. . . . . . . . . . . 12 |- ((((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) /\ x e. A) -> (`'`'2nd |` (({x} X. _V) i^i dom F)) = (2nd |` (({x} X. _V) i^i dom F)))
28	27	rneqd 4188	. . . . . . . . . . 11 |- ((((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) /\ x e. A) -> ran (`'`'2nd |` (({x} X. _V) i^i dom F)) = ran (2nd |` (({x} X. _V) i^i dom F)))
29		fndm 4512	. . . . . . . . . . . . . . 15 |- (F Fn (A X. B) -> dom F = (A X. B))
30		simpr 350	. . . . . . . . . . . . . . . . . . . . 21 |- (((x e. A /\ (A e. C /\ B e. D)) /\ dom F = (A X. B)) -> dom F = (A X. B))
31	30	ineq2d 2796	. . . . . . . . . . . . . . . . . . . 20 |- (((x e. A /\ (A e. C /\ B e. D)) /\ dom F = (A X. B)) -> (({x} X. _V) i^i dom F) = (({x} X. _V) i^i (A X. B)))
32		reseq2 4219	. . . . . . . . . . . . . . . . . . . 20 |- ((({x} X. _V) i^i dom F) = (({x} X. _V) i^i (A X. B)) -> (2nd |` (({x} X. _V) i^i dom F)) = (2nd |` (({x} X. _V) i^i (A X. B))))
33	31, 32	syl 12	. . . . . . . . . . . . . . . . . . 19 |- (((x e. A /\ (A e. C /\ B e. D)) /\ dom F = (A X. B)) -> (2nd |` (({x} X. _V) i^i dom F)) = (2nd |` (({x} X. _V) i^i (A X. B))))
34	33	rneqd 4188	. . . . . . . . . . . . . . . . . 18 |- (((x e. A /\ (A e. C /\ B e. D)) /\ dom F = (A X. B)) -> ran (2nd |` (({x} X. _V) i^i dom F)) = ran (2nd |` (({x} X. _V) i^i (A X. B))))
35		inxp 4109	. . . . . . . . . . . . . . . . . . . . . 22 |- (({x} X. _V) i^i (A X. B)) = (({x} i^i A) X. (_V i^i B))
36		reseq2 4219	. . . . . . . . . . . . . . . . . . . . . 22 |- ((({x} X. _V) i^i (A X. B)) = (({x} i^i A) X. (_V i^i B)) -> (2nd |` (({x} X. _V) i^i (A X. B))) = (2nd |` (({x} i^i A) X. (_V i^i B))))
37	35, 36	ax-mp 7	. . . . . . . . . . . . . . . . . . . . 21 |- (2nd |` (({x} X. _V) i^i (A X. B))) = (2nd |` (({x} i^i A) X. (_V i^i B)))
38	37	rneqi 4187	. . . . . . . . . . . . . . . . . . . 20 |- ran (2nd |` (({x} X. _V) i^i (A X. B))) = ran (2nd |` (({x} i^i A) X. (_V i^i B)))
39	38	a1i 8	. . . . . . . . . . . . . . . . . . 19 |- (((x e. A /\ (A e. C /\ B e. D)) /\ dom F = (A X. B)) -> ran (2nd |` (({x} X. _V) i^i (A X. B))) = ran (2nd |` (({x} i^i A) X. (_V i^i B))))
40		xpeq12 4020	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((({x} i^i A) = {x} /\ (_V i^i B) = B) -> (({x} i^i A) X. (_V i^i B)) = ({x} X. B))
41		snssi 3129	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (x e. A -> {x} C_ A)
42		df-ss 2605	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ({x} C_ A <-> ({x} i^i A) = {x})
43	41, 42	sylib 215	. . . . . . . . . . . . . . . . . . . . . . 23 |- (x e. A -> ({x} i^i A) = {x})
44		incom 2787	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (_V i^i B) = (B i^i _V)
45		inv1 2898	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (B i^i _V) = B
46	44, 45	eqtri 1908	. . . . . . . . . . . . . . . . . . . . . . 23 |- (_V i^i B) = B
47	40, 43, 46	sylancl 525	. . . . . . . . . . . . . . . . . . . . . 22 |- (x e. A -> (({x} i^i A) X. (_V i^i B)) = ({x} X. B))
48		reseq2 4219	. . . . . . . . . . . . . . . . . . . . . 22 |- ((({x} i^i A) X. (_V i^i B)) = ({x} X. B) -> (2nd |` (({x} i^i A) X. (_V i^i B))) = (2nd |` ({x} X. B)))
49	47, 48	syl 12	. . . . . . . . . . . . . . . . . . . . 21 |- (x e. A -> (2nd |` (({x} i^i A) X. (_V i^i B))) = (2nd |` ({x} X. B)))
50	49	rneqd 4188	. . . . . . . . . . . . . . . . . . . 20 |- (x e. A -> ran (2nd |` (({x} i^i A) X. (_V i^i B))) = ran (2nd |` ({x} X. B)))
51	50	ad2antrr 440	. . . . . . . . . . . . . . . . . . 19 |- (((x e. A /\ (A e. C /\ B e. D)) /\ dom F = (A X. B)) -> ran (2nd |` (({x} i^i A) X. (_V i^i B))) = ran (2nd |` ({x} X. B)))
52		2ndconst 5071	. . . . . . . . . . . . . . . . . . . . . 22 |- (x e. _V -> (2nd |` ({x} X. B)):({x} X. B)-1-1-onto->B)
53		pm2.27 76	. . . . . . . . . . . . . . . . . . . . . . 23 |- (x e. _V -> ((x e. _V -> (2nd |` ({x} X. B)):({x} X. B)-1-1-onto->B) -> (2nd |` ({x} X. B)):({x} X. B)-1-1-onto->B))
54		f1ofo 4643	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((2nd |` ({x} X. B)):({x} X. B)-1-1-onto->B -> (2nd |` ({x} X. B)):({x} X. B)-onto->B)
55	53, 54	syl6 25	. . . . . . . . . . . . . . . . . . . . . 22 |- (x e. _V -> ((x e. _V -> (2nd |` ({x} X. B)):({x} X. B)-1-1-onto->B) -> (2nd |` ({x} X. B)):({x} X. B)-onto->B))
56	2, 52, 55	mp2 54	. . . . . . . . . . . . . . . . . . . . 21 |- (2nd |` ({x} X. B)):({x} X. B)-onto->B
57	56	a1i 8	. . . . . . . . . . . . . . . . . . . 20 |- (((x e. A /\ (A e. C /\ B e. D)) /\ dom F = (A X. B)) -> (2nd |` ({x} X. B)):({x} X. B)-onto->B)
58		forn 4620	. . . . . . . . . . . . . . . . . . . 20 |- ((2nd |` ({x} X. B)):({x} X. B)-onto->B -> ran (2nd |` ({x} X. B)) = B)
59	57, 58	syl 12	. . . . . . . . . . . . . . . . . . 19 |- (((x e. A /\ (A e. C /\ B e. D)) /\ dom F = (A X. B)) -> ran (2nd |` ({x} X. B)) = B)
60	39, 51, 59	3eqtrd 1929	. . . . . . . . . . . . . . . . . 18 |- (((x e. A /\ (A e. C /\ B e. D)) /\ dom F = (A X. B)) -> ran (2nd |` (({x} X. _V) i^i (A X. B))) = B)
61	34, 60	eqtrd 1925	. . . . . . . . . . . . . . . . 17 |- (((x e. A /\ (A e. C /\ B e. D)) /\ dom F = (A X. B)) -> ran (2nd |` (({x} X. _V) i^i dom F)) = B)
62	61	exp31 407	. . . . . . . . . . . . . . . 16 |- (x e. A -> ((A e. C /\ B e. D) -> (dom F = (A X. B) -> ran (2nd |` (({x} X. _V) i^i dom F)) = B)))
63	62	com13 37	. . . . . . . . . . . . . . 15 |- (dom F = (A X. B) -> ((A e. C /\ B e. D) -> (x e. A -> ran (2nd |` (({x} X. _V) i^i dom F)) = B)))
64	29, 63	syl 12	. . . . . . . . . . . . . 14 |- (F Fn (A X. B) -> ((A e. C /\ B e. D) -> (x e. A -> ran (2nd |` (({x} X. _V) i^i dom F)) = B)))
65	64	adantl 424	. . . . . . . . . . . . 13 |- ((B =/= (/) /\ F Fn (A X. B)) -> ((A e. C /\ B e. D) -> (x e. A -> ran (2nd |` (({x} X. _V) i^i dom F)) = B)))
66	65	impcom 378	. . . . . . . . . . . 12 |- (((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) -> (x e. A -> ran (2nd |` (({x} X. _V) i^i dom F)) = B))
67	66	imp 377	. . . . . . . . . . 11 |- ((((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) /\ x e. A) -> ran (2nd |` (({x} X. _V) i^i dom F)) = B)
68	28, 67	eqtrd 1925	. . . . . . . . . 10 |- ((((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) /\ x e. A) -> ran (`'`'2nd |` (({x} X. _V) i^i dom F)) = B)
69	22, 25, 68	3eqtrd 1929	. . . . . . . . 9 |- ((((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) /\ x e. A) -> ((`'`'2nd |` ({x} X. _V))"dom F) = B)
70	17, 20, 69	3eqtrd 1929	. . . . . . . 8 |- ((((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) /\ x e. A) -> dom ( F o. `'(2nd |` ({x} X. _V))) = B)
71	15, 70	jca 310	. . . . . . 7 |- ((((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) /\ x e. A) -> (Fun (F o. `'(2nd |` ({x} X. _V))) /\ dom ( F o. `'(2nd |` ({x} X. _V))) = B))
72		df-fn 4009	. . . . . . 7 |- ((F o. `'(2nd |` ({x} X. _V))) Fn B <-> (Fun (F o. `'(2nd |` ({x} X. _V))) /\ dom ( F o. `'(2nd |` ({x} X. _V))) = B))
73	71, 72	sylibr 217	. . . . . 6 |- ((((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) /\ x e. A) -> (F o. `'(2nd |` ({x} X. _V))) Fn B)
74		dffn3 4570	. . . . . 6 |- ((F o. `'(2nd |` ({x} X. _V))) Fn B <-> (F o. `'(2nd |` ({x} X. _V))):B-->ran ( F o. `'(2nd |` ({x} X. _V))))
75	73, 74	sylib 215	. . . . 5 |- ((((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) /\ x e. A) -> (F o. `'(2nd |` ({x} X. _V))):B-->ran ( F o. `'(2nd |` ({x} X. _V))))
76		rncoss 4213	. . . . 5 |- ran ( F o. `'(2nd |` ({x} X. _V))) C_ ran F
77	1, 75, 76	sylancl 525	. . . 4 |- ((((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) /\ x e. A) -> (F o. `'(2nd |` ({x} X. _V))):B-->ran F)
78		fnex 4535	. . . . . . . . . . 11 |- ((F Fn (A X. B) /\ (A X. B) e. _V) -> F e. _V)
79	78	ex 402	. . . . . . . . . 10 |- (F Fn (A X. B) -> ((A X. B) e. _V -> F e. _V))
80		xpexg 4095	. . . . . . . . . 10 |- ((A e. C /\ B e. D) -> (A X. B) e. _V)
81	79, 80	syl5 20	. . . . . . . . 9 |- (F Fn (A X. B) -> ((A e. C /\ B e. D) -> F e. _V))
82	81	adantl 424	. . . . . . . 8 |- ((B =/= (/) /\ F Fn (A X. B)) -> ((A e. C /\ B e. D) -> F e. _V))
83	82	impcom 378	. . . . . . 7 |- (((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) -> F e. _V)
84	83	adantr 425	. . . . . 6 |- ((((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) /\ x e. A) -> F e. _V)
85		fo2nd 5033	. . . . . . . . . 10 |- 2nd:_V-onto->_V
86		fofun 4618	. . . . . . . . . 10 |- (2nd:_V-onto->_V -> Fun 2nd)
87	85, 86	ax-mp 7	. . . . . . . . 9 |- Fun 2nd
88		funres 4459	. . . . . . . . 9 |- (Fun 2nd -> Fun (2nd |` ({x} X. _V)))
89	87, 88	ax-mp 7	. . . . . . . 8 |- Fun (2nd |` ({x} X. _V))
90	89	a1i 8	. . . . . . 7 |- ((((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) /\ x e. A) -> Fun (2nd |` ({x} X. _V)))
91		funcnvcnv 4473	. . . . . . 7 |- (Fun (2nd |` ({x} X. _V)) -> Fun `'`'(2nd |` ({x} X. _V)))
92	90, 91	syl 12	. . . . . 6 |- ((((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) /\ x e. A) -> Fun `'`'(2nd |` ({x} X. _V)))
93		cofunex2g 4502	. . . . . 6 |- ((F e. _V /\ Fun `'`'(2nd |` ({x} X. _V))) -> (F o. `'(2nd |` ({x} X. _V))) e. _V)
94	84, 92, 93	syl11anc 524	. . . . 5 |- ((((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) /\ x e. A) -> (F o. `'(2nd |` ({x} X. _V))) e. _V)
95		feq1 4551	. . . . . 6 |- (f = (F o. `'(2nd |` ({x} X. _V))) -> (f:B-->ran F <-> (F o. `'(2nd |` ({x} X. _V))):B-->ran F))
96	95	elabg 2405	. . . . 5 |- ((F o. `'(2nd |` ({x} X. _V))) e. _V -> ((F o. `'(2nd |` ({x} X. _V))) e. {f | f:B-->ran F} <-> (F o. `'(2nd |` ({x} X. _V))):B-->ran F))
97	94, 96	syl 12	. . . 4 |- ((((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) /\ x e. A) -> ((F o. `'(2nd |` ({x} X. _V))) e. {f | f:B-->ran F} <-> (F o. `'(2nd |` ({x} X. _V))):B-->ran F))
98	77, 97	mpbird 213	. . 3 |- ((((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) /\ x e. A) -> (F o. `'(2nd |` ({x} X. _V))) e. {f | f:B-->ran F})
99	98	r19.21aiva 2176	. 2 |- (((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) -> A.x e. A (F o. `'(2nd |` ({x} X. _V))) e. {f | f:B-->ran F})
100		pm3.22 486	. . . . 5 |- ((B =/= (/) /\ F Fn (A X. B)) -> (F Fn (A X. B) /\ B =/= (/)))
101	100	adantl 424	. . . 4 |- (((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) -> (F Fn (A X. B) /\ B =/= (/)))
102		simpl 346	. . . 4 |- (((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) -> (A e. C /\ B e. D))
103	101, 102	jca 310	. . 3 |- (((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) -> ((F Fn (A X. B) /\ B =/= (/)) /\ (A e. C /\ B e. D)))
104		cur1vald 14547	. . 3 |- (((F Fn (A X. B) /\ B =/= (/)) /\ (A e. C /\ B e. D)) -> (cur1` F) = {<.x, y>. | (x e. A /\ y = (F o. `'(2nd |` ({x} X. _V))))})
105		fopab2g 14485	. . 3 |- ((cur1` F) = {<.x, y>. | (x e. A /\ y = (F o. `'(2nd |` ({x} X. _V))))} -> (A.x e. A (F o. `'(2nd |` ({x} X. _V))) e. {f | f:B-->ran F} <-> (cur1` F):A-->{f | f:B-->ran F}))
106	103, 104, 105	3syl 24	. 2 |- (((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) -> (A.x e. A (F o. `'(2nd |` ({x} X. _V))) e. {f | f:B-->ran F} <-> (cur1` F):A-->{f | f:B-->ran F}))
107	99, 106	mpbid 212	1 |- (((A e. C /\ B e. D) /\ (B =/= (/) /\ F Fn (A X. B))) -> (cur1` F):A-->{f | f:B-->ran F})

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871   =/= wne 2017  A.wral 2105  _Vcvv 2292   i^i cin 2592   C_ wss 2593  (/)c0 2875  {csn 3044  {copab 3395   X. cxp 3984  `'ccnv 3985  dom cdm 3986  ran crn 3987   |` cres 3988  "cima 3989   o. ccom 3990  Fun wfun 3992   Fn wfn 3993  -->wf 3994  -onto->wfo 3996  -1-1-onto->wf1o 3997  ` cfv 3998  2ndc2nd 5019  cur1ccur1 14542

This theorem is referenced by:  valcurfn 14551  curgrpact 14735

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-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-1st 5020  df-2nd 5021  df-cur1 14544

Copyright terms: Public domain