Theorem indf1ofs 30088

Description: The bijection between finite subsets and the indicator functions with finite support. (Contributed by Thierry Arnoux, 22-Aug-2017.)

Assertion

Ref	Expression
indf1ofs	⊢ (𝑂 ∈ 𝑉 → ((𝟭‘𝑂) ↾ Fin):(𝒫 𝑂 ∩ Fin)–1-1-onto→{𝑓 ∈ ({0, 1} ↑𝑚 𝑂) ∣ (◡𝑓 “ {1}) ∈ Fin})

Distinct variable group:   𝑓,𝑂

Allowed substitution hint:   𝑉(𝑓)

Proof of Theorem indf1ofs

Dummy variables 𝑎 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		indf1o 30086	. . . 4 ⊢ (𝑂 ∈ 𝑉 → (𝟭‘𝑂):𝒫 𝑂–1-1-onto→({0, 1} ↑𝑚 𝑂))
2		f1of1 6136	. . . 4 ⊢ ((𝟭‘𝑂):𝒫 𝑂–1-1-onto→({0, 1} ↑𝑚 𝑂) → (𝟭‘𝑂):𝒫 𝑂–1-1→({0, 1} ↑𝑚 𝑂))
3	1, 2	syl 17	. . 3 ⊢ (𝑂 ∈ 𝑉 → (𝟭‘𝑂):𝒫 𝑂–1-1→({0, 1} ↑𝑚 𝑂))
4		inss1 3833	. . 3 ⊢ (𝒫 𝑂 ∩ Fin) ⊆ 𝒫 𝑂
5		f1ores 6151	. . 3 ⊢ (((𝟭‘𝑂):𝒫 𝑂–1-1→({0, 1} ↑𝑚 𝑂) ∧ (𝒫 𝑂 ∩ Fin) ⊆ 𝒫 𝑂) → ((𝟭‘𝑂) ↾ (𝒫 𝑂 ∩ Fin)):(𝒫 𝑂 ∩ Fin)–1-1-onto→((𝟭‘𝑂) “ (𝒫 𝑂 ∩ Fin)))
6	3, 4, 5	sylancl 694	. 2 ⊢ (𝑂 ∈ 𝑉 → ((𝟭‘𝑂) ↾ (𝒫 𝑂 ∩ Fin)):(𝒫 𝑂 ∩ Fin)–1-1-onto→((𝟭‘𝑂) “ (𝒫 𝑂 ∩ Fin)))
7		resres 5409	. . . 4 ⊢ (((𝟭‘𝑂) ↾ 𝒫 𝑂) ↾ Fin) = ((𝟭‘𝑂) ↾ (𝒫 𝑂 ∩ Fin))
8		f1ofn 6138	. . . . . 6 ⊢ ((𝟭‘𝑂):𝒫 𝑂–1-1-onto→({0, 1} ↑𝑚 𝑂) → (𝟭‘𝑂) Fn 𝒫 𝑂)
9		fnresdm 6000	. . . . . 6 ⊢ ((𝟭‘𝑂) Fn 𝒫 𝑂 → ((𝟭‘𝑂) ↾ 𝒫 𝑂) = (𝟭‘𝑂))
10	1, 8, 9	3syl 18	. . . . 5 ⊢ (𝑂 ∈ 𝑉 → ((𝟭‘𝑂) ↾ 𝒫 𝑂) = (𝟭‘𝑂))
11	10	reseq1d 5395	. . . 4 ⊢ (𝑂 ∈ 𝑉 → (((𝟭‘𝑂) ↾ 𝒫 𝑂) ↾ Fin) = ((𝟭‘𝑂) ↾ Fin))
12	7, 11	syl5eqr 2670	. . 3 ⊢ (𝑂 ∈ 𝑉 → ((𝟭‘𝑂) ↾ (𝒫 𝑂 ∩ Fin)) = ((𝟭‘𝑂) ↾ Fin))
13		eqidd 2623	. . 3 ⊢ (𝑂 ∈ 𝑉 → (𝒫 𝑂 ∩ Fin) = (𝒫 𝑂 ∩ Fin))
14		simpll 790	. . . . . . . . . 10 ⊢ (((𝑂 ∈ 𝑉 ∧ 𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → 𝑂 ∈ 𝑉)
15		simpr 477	. . . . . . . . . . . . . . 15 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑎 ∈ (𝒫 𝑂 ∩ Fin)) → 𝑎 ∈ (𝒫 𝑂 ∩ Fin))
16	4, 15	sseldi 3601	. . . . . . . . . . . . . 14 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑎 ∈ (𝒫 𝑂 ∩ Fin)) → 𝑎 ∈ 𝒫 𝑂)
17	16	elpwid 4170	. . . . . . . . . . . . 13 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑎 ∈ (𝒫 𝑂 ∩ Fin)) → 𝑎 ⊆ 𝑂)
18		indf 30077	. . . . . . . . . . . . 13 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑎 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝑎):𝑂⟶{0, 1})
19	17, 18	syldan 487	. . . . . . . . . . . 12 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑎 ∈ (𝒫 𝑂 ∩ Fin)) → ((𝟭‘𝑂)‘𝑎):𝑂⟶{0, 1})
20	19	adantr 481	. . . . . . . . . . 11 ⊢ (((𝑂 ∈ 𝑉 ∧ 𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → ((𝟭‘𝑂)‘𝑎):𝑂⟶{0, 1})
21		simpr 477	. . . . . . . . . . . 12 ⊢ (((𝑂 ∈ 𝑉 ∧ 𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → ((𝟭‘𝑂)‘𝑎) = 𝑔)
22	21	feq1d 6030	. . . . . . . . . . 11 ⊢ (((𝑂 ∈ 𝑉 ∧ 𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → (((𝟭‘𝑂)‘𝑎):𝑂⟶{0, 1} ↔ 𝑔:𝑂⟶{0, 1}))
23	20, 22	mpbid 222	. . . . . . . . . 10 ⊢ (((𝑂 ∈ 𝑉 ∧ 𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → 𝑔:𝑂⟶{0, 1})
24		prex 4909	. . . . . . . . . . . 12 ⊢ {0, 1} ∈ V
25		elmapg 7870	. . . . . . . . . . . 12 ⊢ (({0, 1} ∈ V ∧ 𝑂 ∈ 𝑉) → (𝑔 ∈ ({0, 1} ↑𝑚 𝑂) ↔ 𝑔:𝑂⟶{0, 1}))
26	24, 25	mpan 706	. . . . . . . . . . 11 ⊢ (𝑂 ∈ 𝑉 → (𝑔 ∈ ({0, 1} ↑𝑚 𝑂) ↔ 𝑔:𝑂⟶{0, 1}))
27	26	biimpar 502	. . . . . . . . . 10 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑔:𝑂⟶{0, 1}) → 𝑔 ∈ ({0, 1} ↑𝑚 𝑂))
28	14, 23, 27	syl2anc 693	. . . . . . . . 9 ⊢ (((𝑂 ∈ 𝑉 ∧ 𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → 𝑔 ∈ ({0, 1} ↑𝑚 𝑂))
29	21	cnveqd 5298	. . . . . . . . . . 11 ⊢ (((𝑂 ∈ 𝑉 ∧ 𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → ◡((𝟭‘𝑂)‘𝑎) = ◡𝑔)
30	29	imaeq1d 5465	. . . . . . . . . 10 ⊢ (((𝑂 ∈ 𝑉 ∧ 𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → (◡((𝟭‘𝑂)‘𝑎) “ {1}) = (◡𝑔 “ {1}))
31		indpi1 30082	. . . . . . . . . . . . 13 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑎 ⊆ 𝑂) → (◡((𝟭‘𝑂)‘𝑎) “ {1}) = 𝑎)
32	17, 31	syldan 487	. . . . . . . . . . . 12 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑎 ∈ (𝒫 𝑂 ∩ Fin)) → (◡((𝟭‘𝑂)‘𝑎) “ {1}) = 𝑎)
33		inss2 3834	. . . . . . . . . . . . 13 ⊢ (𝒫 𝑂 ∩ Fin) ⊆ Fin
34	33, 15	sseldi 3601	. . . . . . . . . . . 12 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑎 ∈ (𝒫 𝑂 ∩ Fin)) → 𝑎 ∈ Fin)
35	32, 34	eqeltrd 2701	. . . . . . . . . . 11 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑎 ∈ (𝒫 𝑂 ∩ Fin)) → (◡((𝟭‘𝑂)‘𝑎) “ {1}) ∈ Fin)
36	35	adantr 481	. . . . . . . . . 10 ⊢ (((𝑂 ∈ 𝑉 ∧ 𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → (◡((𝟭‘𝑂)‘𝑎) “ {1}) ∈ Fin)
37	30, 36	eqeltrrd 2702	. . . . . . . . 9 ⊢ (((𝑂 ∈ 𝑉 ∧ 𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → (◡𝑔 “ {1}) ∈ Fin)
38	28, 37	jca 554	. . . . . . . 8 ⊢ (((𝑂 ∈ 𝑉 ∧ 𝑎 ∈ (𝒫 𝑂 ∩ Fin)) ∧ ((𝟭‘𝑂)‘𝑎) = 𝑔) → (𝑔 ∈ ({0, 1} ↑𝑚 𝑂) ∧ (◡𝑔 “ {1}) ∈ Fin))
39	38	ex 450	. . . . . . 7 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑎 ∈ (𝒫 𝑂 ∩ Fin)) → (((𝟭‘𝑂)‘𝑎) = 𝑔 → (𝑔 ∈ ({0, 1} ↑𝑚 𝑂) ∧ (◡𝑔 “ {1}) ∈ Fin)))
40	39	rexlimdva 3031	. . . . . 6 ⊢ (𝑂 ∈ 𝑉 → (∃𝑎 ∈ (𝒫 𝑂 ∩ Fin)((𝟭‘𝑂)‘𝑎) = 𝑔 → (𝑔 ∈ ({0, 1} ↑𝑚 𝑂) ∧ (◡𝑔 “ {1}) ∈ Fin)))
41		cnvimass 5485	. . . . . . . . . 10 ⊢ (◡𝑔 “ {1}) ⊆ dom 𝑔
42	26	biimpa 501	. . . . . . . . . . . 12 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑𝑚 𝑂)) → 𝑔:𝑂⟶{0, 1})
43		fdm 6051	. . . . . . . . . . . 12 ⊢ (𝑔:𝑂⟶{0, 1} → dom 𝑔 = 𝑂)
44	42, 43	syl 17	. . . . . . . . . . 11 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑𝑚 𝑂)) → dom 𝑔 = 𝑂)
45	44	adantrr 753	. . . . . . . . . 10 ⊢ ((𝑂 ∈ 𝑉 ∧ (𝑔 ∈ ({0, 1} ↑𝑚 𝑂) ∧ (◡𝑔 “ {1}) ∈ Fin)) → dom 𝑔 = 𝑂)
46	41, 45	syl5sseq 3653	. . . . . . . . 9 ⊢ ((𝑂 ∈ 𝑉 ∧ (𝑔 ∈ ({0, 1} ↑𝑚 𝑂) ∧ (◡𝑔 “ {1}) ∈ Fin)) → (◡𝑔 “ {1}) ⊆ 𝑂)
47		simprr 796	. . . . . . . . 9 ⊢ ((𝑂 ∈ 𝑉 ∧ (𝑔 ∈ ({0, 1} ↑𝑚 𝑂) ∧ (◡𝑔 “ {1}) ∈ Fin)) → (◡𝑔 “ {1}) ∈ Fin)
48		elfpw 8268	. . . . . . . . 9 ⊢ ((◡𝑔 “ {1}) ∈ (𝒫 𝑂 ∩ Fin) ↔ ((◡𝑔 “ {1}) ⊆ 𝑂 ∧ (◡𝑔 “ {1}) ∈ Fin))
49	46, 47, 48	sylanbrc 698	. . . . . . . 8 ⊢ ((𝑂 ∈ 𝑉 ∧ (𝑔 ∈ ({0, 1} ↑𝑚 𝑂) ∧ (◡𝑔 “ {1}) ∈ Fin)) → (◡𝑔 “ {1}) ∈ (𝒫 𝑂 ∩ Fin))
50		indpreima 30087	. . . . . . . . . . 11 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑔:𝑂⟶{0, 1}) → 𝑔 = ((𝟭‘𝑂)‘(◡𝑔 “ {1})))
51	50	eqcomd 2628	. . . . . . . . . 10 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑔:𝑂⟶{0, 1}) → ((𝟭‘𝑂)‘(◡𝑔 “ {1})) = 𝑔)
52	42, 51	syldan 487	. . . . . . . . 9 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝑔 ∈ ({0, 1} ↑𝑚 𝑂)) → ((𝟭‘𝑂)‘(◡𝑔 “ {1})) = 𝑔)
53	52	adantrr 753	. . . . . . . 8 ⊢ ((𝑂 ∈ 𝑉 ∧ (𝑔 ∈ ({0, 1} ↑𝑚 𝑂) ∧ (◡𝑔 “ {1}) ∈ Fin)) → ((𝟭‘𝑂)‘(◡𝑔 “ {1})) = 𝑔)
54		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑎 = (◡𝑔 “ {1}) → ((𝟭‘𝑂)‘𝑎) = ((𝟭‘𝑂)‘(◡𝑔 “ {1})))
55	54	eqeq1d 2624	. . . . . . . . 9 ⊢ (𝑎 = (◡𝑔 “ {1}) → (((𝟭‘𝑂)‘𝑎) = 𝑔 ↔ ((𝟭‘𝑂)‘(◡𝑔 “ {1})) = 𝑔))
56	55	rspcev 3309	. . . . . . . 8 ⊢ (((◡𝑔 “ {1}) ∈ (𝒫 𝑂 ∩ Fin) ∧ ((𝟭‘𝑂)‘(◡𝑔 “ {1})) = 𝑔) → ∃𝑎 ∈ (𝒫 𝑂 ∩ Fin)((𝟭‘𝑂)‘𝑎) = 𝑔)
57	49, 53, 56	syl2anc 693	. . . . . . 7 ⊢ ((𝑂 ∈ 𝑉 ∧ (𝑔 ∈ ({0, 1} ↑𝑚 𝑂) ∧ (◡𝑔 “ {1}) ∈ Fin)) → ∃𝑎 ∈ (𝒫 𝑂 ∩ Fin)((𝟭‘𝑂)‘𝑎) = 𝑔)
58	57	ex 450	. . . . . 6 ⊢ (𝑂 ∈ 𝑉 → ((𝑔 ∈ ({0, 1} ↑𝑚 𝑂) ∧ (◡𝑔 “ {1}) ∈ Fin) → ∃𝑎 ∈ (𝒫 𝑂 ∩ Fin)((𝟭‘𝑂)‘𝑎) = 𝑔))
59	40, 58	impbid 202	. . . . 5 ⊢ (𝑂 ∈ 𝑉 → (∃𝑎 ∈ (𝒫 𝑂 ∩ Fin)((𝟭‘𝑂)‘𝑎) = 𝑔 ↔ (𝑔 ∈ ({0, 1} ↑𝑚 𝑂) ∧ (◡𝑔 “ {1}) ∈ Fin)))
60	1, 8	syl 17	. . . . . 6 ⊢ (𝑂 ∈ 𝑉 → (𝟭‘𝑂) Fn 𝒫 𝑂)
61		fvelimab 6253	. . . . . 6 ⊢ (((𝟭‘𝑂) Fn 𝒫 𝑂 ∧ (𝒫 𝑂 ∩ Fin) ⊆ 𝒫 𝑂) → (𝑔 ∈ ((𝟭‘𝑂) “ (𝒫 𝑂 ∩ Fin)) ↔ ∃𝑎 ∈ (𝒫 𝑂 ∩ Fin)((𝟭‘𝑂)‘𝑎) = 𝑔))
62	60, 4, 61	sylancl 694	. . . . 5 ⊢ (𝑂 ∈ 𝑉 → (𝑔 ∈ ((𝟭‘𝑂) “ (𝒫 𝑂 ∩ Fin)) ↔ ∃𝑎 ∈ (𝒫 𝑂 ∩ Fin)((𝟭‘𝑂)‘𝑎) = 𝑔))
63		cnveq 5296	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → ◡𝑓 = ◡𝑔)
64	63	imaeq1d 5465	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (◡𝑓 “ {1}) = (◡𝑔 “ {1}))
65	64	eleq1d 2686	. . . . . . 7 ⊢ (𝑓 = 𝑔 → ((◡𝑓 “ {1}) ∈ Fin ↔ (◡𝑔 “ {1}) ∈ Fin))
66	65	elrab 3363	. . . . . 6 ⊢ (𝑔 ∈ {𝑓 ∈ ({0, 1} ↑𝑚 𝑂) ∣ (◡𝑓 “ {1}) ∈ Fin} ↔ (𝑔 ∈ ({0, 1} ↑𝑚 𝑂) ∧ (◡𝑔 “ {1}) ∈ Fin))
67	66	a1i 11	. . . . 5 ⊢ (𝑂 ∈ 𝑉 → (𝑔 ∈ {𝑓 ∈ ({0, 1} ↑𝑚 𝑂) ∣ (◡𝑓 “ {1}) ∈ Fin} ↔ (𝑔 ∈ ({0, 1} ↑𝑚 𝑂) ∧ (◡𝑔 “ {1}) ∈ Fin)))
68	59, 62, 67	3bitr4d 300	. . . 4 ⊢ (𝑂 ∈ 𝑉 → (𝑔 ∈ ((𝟭‘𝑂) “ (𝒫 𝑂 ∩ Fin)) ↔ 𝑔 ∈ {𝑓 ∈ ({0, 1} ↑𝑚 𝑂) ∣ (◡𝑓 “ {1}) ∈ Fin}))
69	68	eqrdv 2620	. . 3 ⊢ (𝑂 ∈ 𝑉 → ((𝟭‘𝑂) “ (𝒫 𝑂 ∩ Fin)) = {𝑓 ∈ ({0, 1} ↑𝑚 𝑂) ∣ (◡𝑓 “ {1}) ∈ Fin})
70	12, 13, 69	f1oeq123d 6133	. 2 ⊢ (𝑂 ∈ 𝑉 → (((𝟭‘𝑂) ↾ (𝒫 𝑂 ∩ Fin)):(𝒫 𝑂 ∩ Fin)–1-1-onto→((𝟭‘𝑂) “ (𝒫 𝑂 ∩ Fin)) ↔ ((𝟭‘𝑂) ↾ Fin):(𝒫 𝑂 ∩ Fin)–1-1-onto→{𝑓 ∈ ({0, 1} ↑𝑚 𝑂) ∣ (◡𝑓 “ {1}) ∈ Fin}))
71	6, 70	mpbid 222	1 ⊢ (𝑂 ∈ 𝑉 → ((𝟭‘𝑂) ↾ Fin):(𝒫 𝑂 ∩ Fin)–1-1-onto→{𝑓 ∈ ({0, 1} ↑𝑚 𝑂) ∣ (◡𝑓 “ {1}) ∈ Fin})

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∃wrex 2913  {crab 2916  Vcvv 3200   ∩ cin 3573   ⊆ wss 3574  𝒫 cpw 4158  {csn 4177  {cpr 4179  ^◡ccnv 5113  dom cdm 5114   ↾ cres 5116   “ cima 5117   Fn wfn 5883  ⟶wf 5884  –1-1→wf1 5885  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857  Fincfn 7955  0cc0 9936  1c1 9937  𝟭cind 30072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-i2m1 10004  ax-1ne0 10005  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-ind 30073

This theorem is referenced by:  eulerpartgbij  30434