Theorem pwfi2f1o 43086

Description: The pw2f1o 9099 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 2734	. . . . 5 ⊢ (𝑥 ∈ (2o ↑m 𝐴) ↦ (◡𝑥 “ {1o})) = (𝑥 ∈ (2o ↑m 𝐴) ↦ (◡𝑥 “ {1o}))
2	1	pw2f1o2 43028	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (2o ↑m 𝐴) ↦ (◡𝑥 “ {1o})):(2o ↑m 𝐴)–1-1-onto→𝒫 𝐴)
3		f1of1 6827	. . . 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 4060	. . . 4 ⊢ {𝑦 ∈ (2o ↑m 𝐴) ∣ 𝑦 finSupp ∅} ⊆ (2o ↑m 𝐴)
7	5, 6	eqsstri 4010	. . 3 ⊢ 𝑆 ⊆ (2o ↑m 𝐴)
8		f1ores 6842	. . 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 586	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (2o ↑m 𝐴) ↦ (◡𝑥 “ {1o})) ↾ 𝑆):𝑆–1-1-onto→((𝑥 ∈ (2o ↑m 𝐴) ↦ (◡𝑥 “ {1o})) “ 𝑆))
10		elmapfun 8888	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (2o ↑m 𝐴) → Fun 𝑦)
11		id 22	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (2o ↑m 𝐴) → 𝑦 ∈ (2o ↑m 𝐴))
12		0ex 5287	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ V
13	12	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (2o ↑m 𝐴) → ∅ ∈ V)
14	10, 11, 13	3jca 1128	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (2o ↑m 𝐴) → (Fun 𝑦 ∧ 𝑦 ∈ (2o ↑m 𝐴) ∧ ∅ ∈ V))
15	14	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2o ↑m 𝐴)) → (Fun 𝑦 ∧ 𝑦 ∈ (2o ↑m 𝐴) ∧ ∅ ∈ V))
16		funisfsupp 9389	. . . . . . . . . . 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 617	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2o ↑m 𝐴)) → (𝐴 ∈ 𝑉 ∧ ∅ ∈ V))
19		elmapi 8871	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (2o ↑m 𝐴) → 𝑦:𝐴⟶2o)
20	19	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2o ↑m 𝐴)) → 𝑦:𝐴⟶2o)
21		fsuppeq 8182	. . . . . . . . . . . . 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 8489	. . . . . . . . . . . . . . . 16 ⊢ 2o = suc 1o
24		df-suc 6369	. . . . . . . . . . . . . . . . 17 ⊢ suc 1o = (1o ∪ {1o})
25	24	equncomi 4140	. . . . . . . . . . . . . . . 16 ⊢ suc 1o = ({1o} ∪ 1o)
26	23, 25	eqtri 2757	. . . . . . . . . . . . . . 15 ⊢ 2o = ({1o} ∪ 1o)
27		df1o2 8495	. . . . . . . . . . . . . . . 16 ⊢ 1o = {∅}
28	27	eqcomi 2743	. . . . . . . . . . . . . . 15 ⊢ {∅} = 1o
29	26, 28	difeq12i 4104	. . . . . . . . . . . . . 14 ⊢ (2o ∖ {∅}) = (({1o} ∪ 1o) ∖ 1o)
30		difun2 4461	. . . . . . . . . . . . . . 15 ⊢ (({1o} ∪ 1o) ∖ 1o) = ({1o} ∖ 1o)
31		incom 4189	. . . . . . . . . . . . . . . . 17 ⊢ ({1o} ∩ 1o) = (1o ∩ {1o})
32		1on 8500	. . . . . . . . . . . . . . . . . . 19 ⊢ 1o ∈ On
33	32	onordi 6475	. . . . . . . . . . . . . . . . . 18 ⊢ Ord 1o
34		orddisj 6401	. . . . . . . . . . . . . . . . . 18 ⊢ (Ord 1o → (1o ∩ {1o}) = ∅)
35	33, 34	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (1o ∩ {1o}) = ∅
36	31, 35	eqtri 2757	. . . . . . . . . . . . . . . 16 ⊢ ({1o} ∩ 1o) = ∅
37		disj3 4434	. . . . . . . . . . . . . . . 16 ⊢ (({1o} ∩ 1o) = ∅ ↔ {1o} = ({1o} ∖ 1o))
38	36, 37	mpbi 230	. . . . . . . . . . . . . . 15 ⊢ {1o} = ({1o} ∖ 1o)
39	30, 38	eqtr4i 2760	. . . . . . . . . . . . . 14 ⊢ (({1o} ∪ 1o) ∖ 1o) = {1o}
40	29, 39	eqtri 2757	. . . . . . . . . . . . 13 ⊢ (2o ∖ {∅}) = {1o}
41	40	imaeq2i 6056	. . . . . . . . . . . 12 ⊢ (◡𝑦 “ (2o ∖ {∅})) = (◡𝑦 “ {1o})
42	22, 41	eqtrdi 2785	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2o ↑m 𝐴)) → (𝑦 supp ∅) = (◡𝑦 “ {1o}))
43	42	eleq1d 2818	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2o ↑m 𝐴)) → ((𝑦 supp ∅) ∈ Fin ↔ (◡𝑦 “ {1o}) ∈ Fin))
44		cnvimass 6080	. . . . . . . . . . . 12 ⊢ (◡𝑦 “ {1o}) ⊆ dom 𝑦
45	44, 20	fssdm 6735	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2o ↑m 𝐴)) → (◡𝑦 “ {1o}) ⊆ 𝐴)
46	45	biantrurd 532	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2o ↑m 𝐴)) → ((◡𝑦 “ {1o}) ∈ Fin ↔ ((◡𝑦 “ {1o}) ⊆ 𝐴 ∧ (◡𝑦 “ {1o}) ∈ Fin)))
47	17, 43, 46	3bitrd 305	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2o ↑m 𝐴)) → (𝑦 finSupp ∅ ↔ ((◡𝑦 “ {1o}) ⊆ 𝐴 ∧ (◡𝑦 “ {1o}) ∈ Fin)))
48		elfpw 9376	. . . . . . . . 9 ⊢ ((◡𝑦 “ {1o}) ∈ (𝒫 𝐴 ∩ Fin) ↔ ((◡𝑦 “ {1o}) ⊆ 𝐴 ∧ (◡𝑦 “ {1o}) ∈ Fin))
49	47, 48	bitr4di 289	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ (2o ↑m 𝐴)) → (𝑦 finSupp ∅ ↔ (◡𝑦 “ {1o}) ∈ (𝒫 𝐴 ∩ Fin)))
50	49	rabbidva 3426	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∈ (2o ↑m 𝐴) ∣ 𝑦 finSupp ∅} = {𝑦 ∈ (2o ↑m 𝐴) ∣ (◡𝑦 “ {1o}) ∈ (𝒫 𝐴 ∩ Fin)})
51		cnveq 5864	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ◡𝑥 = ◡𝑦)
52	51	imaeq1d 6057	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (◡𝑥 “ {1o}) = (◡𝑦 “ {1o}))
53	52	cbvmptv 5235	. . . . . . . 8 ⊢ (𝑥 ∈ (2o ↑m 𝐴) ↦ (◡𝑥 “ {1o})) = (𝑦 ∈ (2o ↑m 𝐴) ↦ (◡𝑦 “ {1o}))
54	53	mptpreima 6238	. . . . . . 7 ⊢ (◡(𝑥 ∈ (2o ↑m 𝐴) ↦ (◡𝑥 “ {1o})) “ (𝒫 𝐴 ∩ Fin)) = {𝑦 ∈ (2o ↑m 𝐴) ∣ (◡𝑦 “ {1o}) ∈ (𝒫 𝐴 ∩ Fin)}
55	50, 5, 54	3eqtr4g 2794	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝑆 = (◡(𝑥 ∈ (2o ↑m 𝐴) ↦ (◡𝑥 “ {1o})) “ (𝒫 𝐴 ∩ Fin)))
56	55	imaeq2d 6058	. . . . 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 6835	. . . . . . 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 4217	. . . . . 6 ⊢ (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴
60		foimacnv 6845	. . . . . 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 586	. . . . 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 2769	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (2o ↑m 𝐴) ↦ (◡𝑥 “ {1o})) “ 𝑆) = (𝒫 𝐴 ∩ Fin))
63		f1oeq3 6818	. . . 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 6035	. . . . . 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 2760	. . . 4 ⊢ ((𝑥 ∈ (2o ↑m 𝐴) ↦ (◡𝑥 “ {1o})) ↾ 𝑆) = 𝐹
69		f1oeq1 6816	. . . 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 279	. 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 232	1 ⊢ (𝐴 ∈ 𝑉 → 𝐹:𝑆–1-1-onto→(𝒫 𝐴 ∩ Fin))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  {crab 3419  Vcvv 3463   ∖ cdif 3928   ∪ cun 3929   ∩ cin 3930   ⊆ wss 3931  ∅c0 4313  𝒫 cpw 4580  {csn 4606   class class class wbr 5123   ↦ cmpt 5205  ^◡ccnv 5664   ↾ cres 5667   “ cima 5668  Ord word 6362  suc csuc 6365  Fun wfun 6535  ⟶wf 6537  –1-1→wf1 6538  –onto→wfo 6539  –1-1-onto→wf1o 6540  (class class class)co 7413   supp csupp 8167  1_oc1o 8481  2_oc2o 8482   ↑_m cmap 8848  Fincfn 8967   finSupp cfsupp 9383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-ord 6366  df-on 6367  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7996  df-2nd 7997  df-supp 8168  df-1o 8488  df-2o 8489  df-map 8850  df-fsupp 9384

This theorem is referenced by:  pwfi2en  43087