Theorem pwfi2f1o 39126

Description: The pw2f1o 8417 bijection relates finitely supported indicator functions on a two-element set to finite subsets. MOVABLE (Contributed by Stefan O'Rear, 10-Jul-2015.) (Revised by AV, 14-Jun-2020.)

Hypotheses

Ref	Expression
pwfi2f1o.s	⊢ 𝑆 = {𝑦 ∈ (2o ↑𝑚 𝐴) ∣ 𝑦 finSupp ∅}
pwfi2f1o.f	⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ (◡𝑥 “ {1o}))

Assertion

Ref	Expression
pwfi2f1o	⊢ (𝐴 ∈ 𝑉 → 𝐹:𝑆–1-1-onto→(𝒫 𝐴 ∩ Fin))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑆   𝑥,𝑉,𝑦

Allowed substitution hints:   𝑆(𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem pwfi2f1o

Step	Hyp	Ref	Expression
1		eqid 2773	. . . . 5 ⊢ (𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})) = (𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o}))
2	1	pw2f1o2 39065	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})):(2o ↑𝑚 𝐴)–1-1-onto→𝒫 𝐴)
3		f1of1 6441	. . . 4 ⊢ ((𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})):(2o ↑𝑚 𝐴)–1-1-onto→𝒫 𝐴 → (𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})):(2o ↑𝑚 𝐴)–1-1→𝒫 𝐴)
4	2, 3	syl 17	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})):(2o ↑𝑚 𝐴)–1-1→𝒫 𝐴)
5		pwfi2f1o.s	. . . 4 ⊢ 𝑆 = {𝑦 ∈ (2o ↑𝑚 𝐴) ∣ 𝑦 finSupp ∅}
6		ssrab2 3941	. . . 4 ⊢ {𝑦 ∈ (2o ↑𝑚 𝐴) ∣ 𝑦 finSupp ∅} ⊆ (2o ↑𝑚 𝐴)
7	5, 6	eqsstri 3886	. . 3 ⊢ 𝑆 ⊆ (2o ↑𝑚 𝐴)
8		f1ores 6456	. . 3 ⊢ (((𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})):(2o ↑𝑚 𝐴)–1-1→𝒫 𝐴 ∧ 𝑆 ⊆ (2o ↑𝑚 𝐴)) → ((𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})) ↾ 𝑆):𝑆–1-1-onto→((𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})) “ 𝑆))
9	4, 7, 8	sylancl 578	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})) ↾ 𝑆):𝑆–1-1-onto→((𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})) “ 𝑆))
10		elmapfun 8229	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (2o ↑𝑚 𝐴) → Fun 𝑦)
11		id 22	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (2o ↑𝑚 𝐴) → 𝑦 ∈ (2o ↑𝑚 𝐴))
12		0ex 5065	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ V
13	12	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (2o ↑𝑚 𝐴) → ∅ ∈ V)
14	10, 11, 13	3jca 1109	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (2o ↑𝑚 𝐴) → (Fun 𝑦 ∧ 𝑦 ∈ (2o ↑𝑚 𝐴) ∧ ∅ ∈ V))
15	14	adantl 474	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2o ↑𝑚 𝐴)) → (Fun 𝑦 ∧ 𝑦 ∈ (2o ↑𝑚 𝐴) ∧ ∅ ∈ V))
16		funisfsupp 8632	. . . . . . . . . . 11 ⊢ ((Fun 𝑦 ∧ 𝑦 ∈ (2o ↑𝑚 𝐴) ∧ ∅ ∈ V) → (𝑦 finSupp ∅ ↔ (𝑦 supp ∅) ∈ Fin))
17	15, 16	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2o ↑𝑚 𝐴)) → (𝑦 finSupp ∅ ↔ (𝑦 supp ∅) ∈ Fin))
18	13	anim2i 608	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2o ↑𝑚 𝐴)) → (𝐴 ∈ 𝑉 ∧ ∅ ∈ V))
19		elmapi 8227	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (2o ↑𝑚 𝐴) → 𝑦:𝐴⟶2o)
20	19	adantl 474	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2o ↑𝑚 𝐴)) → 𝑦:𝐴⟶2o)
21		frnsuppeq 7644	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ ∅ ∈ V) → (𝑦:𝐴⟶2o → (𝑦 supp ∅) = (◡𝑦 “ (2o ∖ {∅}))))
22	18, 20, 21	sylc 65	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2o ↑𝑚 𝐴)) → (𝑦 supp ∅) = (◡𝑦 “ (2o ∖ {∅})))
23		df-2o 7905	. . . . . . . . . . . . . . . 16 ⊢ 2o = suc 1o
24		df-suc 6033	. . . . . . . . . . . . . . . . 17 ⊢ suc 1o = (1o ∪ {1o})
25	24	equncomi 4015	. . . . . . . . . . . . . . . 16 ⊢ suc 1o = ({1o} ∪ 1o)
26	23, 25	eqtri 2797	. . . . . . . . . . . . . . 15 ⊢ 2o = ({1o} ∪ 1o)
27		df1o2 7917	. . . . . . . . . . . . . . . 16 ⊢ 1o = {∅}
28	27	eqcomi 2782	. . . . . . . . . . . . . . 15 ⊢ {∅} = 1o
29	26, 28	difeq12i 3982	. . . . . . . . . . . . . 14 ⊢ (2o ∖ {∅}) = (({1o} ∪ 1o) ∖ 1o)
30		difun2 4307	. . . . . . . . . . . . . . 15 ⊢ (({1o} ∪ 1o) ∖ 1o) = ({1o} ∖ 1o)
31		incom 4061	. . . . . . . . . . . . . . . . 17 ⊢ ({1o} ∩ 1o) = (1o ∩ {1o})
32		1on 7911	. . . . . . . . . . . . . . . . . . 19 ⊢ 1o ∈ On
33	32	onordi 6131	. . . . . . . . . . . . . . . . . 18 ⊢ Ord 1o
34		orddisj 6065	. . . . . . . . . . . . . . . . . 18 ⊢ (Ord 1o → (1o ∩ {1o}) = ∅)
35	33, 34	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (1o ∩ {1o}) = ∅
36	31, 35	eqtri 2797	. . . . . . . . . . . . . . . 16 ⊢ ({1o} ∩ 1o) = ∅
37		disj3 4281	. . . . . . . . . . . . . . . 16 ⊢ (({1o} ∩ 1o) = ∅ ↔ {1o} = ({1o} ∖ 1o))
38	36, 37	mpbi 222	. . . . . . . . . . . . . . 15 ⊢ {1o} = ({1o} ∖ 1o)
39	30, 38	eqtr4i 2800	. . . . . . . . . . . . . 14 ⊢ (({1o} ∪ 1o) ∖ 1o) = {1o}
40	29, 39	eqtri 2797	. . . . . . . . . . . . 13 ⊢ (2o ∖ {∅}) = {1o}
41	40	imaeq2i 5766	. . . . . . . . . . . 12 ⊢ (◡𝑦 “ (2o ∖ {∅})) = (◡𝑦 “ {1o})
42	22, 41	syl6eq 2825	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2o ↑𝑚 𝐴)) → (𝑦 supp ∅) = (◡𝑦 “ {1o}))
43	42	eleq1d 2845	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2o ↑𝑚 𝐴)) → ((𝑦 supp ∅) ∈ Fin ↔ (◡𝑦 “ {1o}) ∈ Fin))
44		cnvimass 5787	. . . . . . . . . . . 12 ⊢ (◡𝑦 “ {1o}) ⊆ dom 𝑦
45	44, 20	fssdm 6358	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2o ↑𝑚 𝐴)) → (◡𝑦 “ {1o}) ⊆ 𝐴)
46	45	biantrurd 525	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2o ↑𝑚 𝐴)) → ((◡𝑦 “ {1o}) ∈ Fin ↔ ((◡𝑦 “ {1o}) ⊆ 𝐴 ∧ (◡𝑦 “ {1o}) ∈ Fin)))
47	17, 43, 46	3bitrd 297	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2o ↑𝑚 𝐴)) → (𝑦 finSupp ∅ ↔ ((◡𝑦 “ {1o}) ⊆ 𝐴 ∧ (◡𝑦 “ {1o}) ∈ Fin)))
48		elfpw 8620	. . . . . . . . 9 ⊢ ((◡𝑦 “ {1o}) ∈ (𝒫 𝐴 ∩ Fin) ↔ ((◡𝑦 “ {1o}) ⊆ 𝐴 ∧ (◡𝑦 “ {1o}) ∈ Fin))
49	47, 48	syl6bbr 281	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2o ↑𝑚 𝐴)) → (𝑦 finSupp ∅ ↔ (◡𝑦 “ {1o}) ∈ (𝒫 𝐴 ∩ Fin)))
50	49	rabbidva 3397	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∈ (2o ↑𝑚 𝐴) ∣ 𝑦 finSupp ∅} = {𝑦 ∈ (2o ↑𝑚 𝐴) ∣ (◡𝑦 “ {1o}) ∈ (𝒫 𝐴 ∩ Fin)})
51		cnveq 5591	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ◡𝑥 = ◡𝑦)
52	51	imaeq1d 5767	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (◡𝑥 “ {1o}) = (◡𝑦 “ {1o}))
53	52	cbvmptv 5025	. . . . . . . 8 ⊢ (𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})) = (𝑦 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑦 “ {1o}))
54	53	mptpreima 5929	. . . . . . 7 ⊢ (◡(𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})) “ (𝒫 𝐴 ∩ Fin)) = {𝑦 ∈ (2o ↑𝑚 𝐴) ∣ (◡𝑦 “ {1o}) ∈ (𝒫 𝐴 ∩ Fin)}
55	50, 5, 54	3eqtr4g 2834	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝑆 = (◡(𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})) “ (𝒫 𝐴 ∩ Fin)))
56	55	imaeq2d 5768	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})) “ 𝑆) = ((𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})) “ (◡(𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})) “ (𝒫 𝐴 ∩ Fin))))
57		f1ofo 6449	. . . . . . 7 ⊢ ((𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})):(2o ↑𝑚 𝐴)–1-1-onto→𝒫 𝐴 → (𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})):(2o ↑𝑚 𝐴)–onto→𝒫 𝐴)
58	2, 57	syl 17	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})):(2o ↑𝑚 𝐴)–onto→𝒫 𝐴)
59		inss1 4087	. . . . . 6 ⊢ (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴
60		foimacnv 6459	. . . . . 6 ⊢ (((𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})):(2o ↑𝑚 𝐴)–onto→𝒫 𝐴 ∧ (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴) → ((𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})) “ (◡(𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})) “ (𝒫 𝐴 ∩ Fin))) = (𝒫 𝐴 ∩ Fin))
61	58, 59, 60	sylancl 578	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})) “ (◡(𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})) “ (𝒫 𝐴 ∩ Fin))) = (𝒫 𝐴 ∩ Fin))
62	56, 61	eqtrd 2809	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})) “ 𝑆) = (𝒫 𝐴 ∩ Fin))
63		f1oeq3 6433	. . . 4 ⊢ (((𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})) “ 𝑆) = (𝒫 𝐴 ∩ Fin) → (((𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})) ↾ 𝑆):𝑆–1-1-onto→((𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})) “ 𝑆) ↔ ((𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})) ↾ 𝑆):𝑆–1-1-onto→(𝒫 𝐴 ∩ Fin)))
64	62, 63	syl 17	. . 3 ⊢ (𝐴 ∈ 𝑉 → (((𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})) ↾ 𝑆):𝑆–1-1-onto→((𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})) “ 𝑆) ↔ ((𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})) ↾ 𝑆):𝑆–1-1-onto→(𝒫 𝐴 ∩ Fin)))
65		resmpt 5748	. . . . . 6 ⊢ (𝑆 ⊆ (2o ↑𝑚 𝐴) → ((𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})) ↾ 𝑆) = (𝑥 ∈ 𝑆 ↦ (◡𝑥 “ {1o})))
66	7, 65	ax-mp 5	. . . . 5 ⊢ ((𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})) ↾ 𝑆) = (𝑥 ∈ 𝑆 ↦ (◡𝑥 “ {1o}))
67		pwfi2f1o.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ (◡𝑥 “ {1o}))
68	66, 67	eqtr4i 2800	. . . 4 ⊢ ((𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})) ↾ 𝑆) = 𝐹
69		f1oeq1 6431	. . . 4 ⊢ (((𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})) ↾ 𝑆) = 𝐹 → (((𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})) ↾ 𝑆):𝑆–1-1-onto→(𝒫 𝐴 ∩ Fin) ↔ 𝐹:𝑆–1-1-onto→(𝒫 𝐴 ∩ Fin)))
70	68, 69	mp1i 13	. . 3 ⊢ (𝐴 ∈ 𝑉 → (((𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})) ↾ 𝑆):𝑆–1-1-onto→(𝒫 𝐴 ∩ Fin) ↔ 𝐹:𝑆–1-1-onto→(𝒫 𝐴 ∩ Fin)))
71	64, 70	bitrd 271	. 2 ⊢ (𝐴 ∈ 𝑉 → (((𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})) ↾ 𝑆):𝑆–1-1-onto→((𝑥 ∈ (2o ↑𝑚 𝐴) ↦ (◡𝑥 “ {1o})) “ 𝑆) ↔ 𝐹:𝑆–1-1-onto→(𝒫 𝐴 ∩ Fin)))
72	9, 71	mpbid 224	1 ⊢ (𝐴 ∈ 𝑉 → 𝐹:𝑆–1-1-onto→(𝒫 𝐴 ∩ Fin))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   ∧ w3a 1069   = wceq 1508   ∈ wcel 2051  {crab 3087  Vcvv 3410   ∖ cdif 3821   ∪ cun 3822   ∩ cin 3823   ⊆ wss 3824  ∅c0 4173  𝒫 cpw 4417  {csn 4436   class class class wbr 4926   ↦ cmpt 5005  ^◡ccnv 5403   ↾ cres 5406   “ cima 5407  Ord word 6026  suc csuc 6029  Fun wfun 6180  ⟶wf 6182  –1-1→wf1 6183  –onto→wfo 6184  –1-1-onto→wf1o 6185  (class class class)co 6975   supp csupp 7632  1_oc1o 7897  2_oc2o 7898   ↑_𝑚 cmap 8205  Fincfn 8305   finSupp cfsupp 8627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-rep 5046  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-ral 3088  df-rex 3089  df-reu 3090  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-pss 3840  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-tp 4441  df-op 4443  df-uni 4710  df-iun 4791  df-br 4927  df-opab 4989  df-mpt 5006  df-tr 5028  df-id 5309  df-eprel 5314  df-po 5323  df-so 5324  df-fr 5363  df-we 5365  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-ord 6030  df-on 6031  df-suc 6033  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-ov 6978  df-oprab 6979  df-mpo 6980  df-1st 7500  df-2nd 7501  df-supp 7633  df-1o 7904  df-2o 7905  df-map 8207  df-fsupp 8628

This theorem is referenced by:  pwfi2en  39127