Theorem pw2f1ocnv 36622

Description: Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 7952, which can be easily reproved in terms of this. (Contributed by Stefan O'Rear, 18-Jan-2015.) (Revised by Stefan O'Rear, 9-Jul-2015.)

Hypothesis

Ref	Expression
pw2f1o2.f	⊢ 𝐹 = (𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜}))

Assertion

Ref	Expression
pw2f1ocnv	⊢ (𝐴 ∈ 𝑉 → (𝐹:(2𝑜 ↑𝑚 𝐴)–1-1-onto→𝒫 𝐴 ∧ ◡𝐹 = (𝑦 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅)))))

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

Allowed substitution hints:   𝐹(𝑥,𝑦,𝑧)   𝑉(𝑧)

Proof of Theorem pw2f1ocnv

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pw2f1o2.f	. 2 ⊢ 𝐹 = (𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑥 “ {1𝑜}))
2		vex 3176	. . . 4 ⊢ 𝑥 ∈ V
3	2	cnvex 7006	. . 3 ⊢ ◡𝑥 ∈ V
4		imaexg 6995	. . 3 ⊢ (◡𝑥 ∈ V → (◡𝑥 “ {1𝑜}) ∈ V)
5	3, 4	mp1i 13	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (2𝑜 ↑𝑚 𝐴)) → (◡𝑥 “ {1𝑜}) ∈ V)
6		mptexg 6389	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅)) ∈ V)
7	6	adantr 480	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝐴) → (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅)) ∈ V)
8		2on 7455	. . . . . 6 ⊢ 2𝑜 ∈ On
9		elmapg 7757	. . . . . 6 ⊢ ((2𝑜 ∈ On ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶2𝑜))
10	8, 9	mpan 702	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ↔ 𝑥:𝐴⟶2𝑜))
11	10	anbi1d 737	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ∧ 𝑦 = (◡𝑥 “ {1𝑜})) ↔ (𝑥:𝐴⟶2𝑜 ∧ 𝑦 = (◡𝑥 “ {1𝑜}))))
12		1on 7454	. . . . . . . . . . . . 13 ⊢ 1𝑜 ∈ On
13	12	elexi 3186	. . . . . . . . . . . 12 ⊢ 1𝑜 ∈ V
14	13	sucid 5721	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ suc 1𝑜
15		df-2o 7448	. . . . . . . . . . 11 ⊢ 2𝑜 = suc 1𝑜
16	14, 15	eleqtrri 2687	. . . . . . . . . 10 ⊢ 1𝑜 ∈ 2𝑜
17		0ex 4718	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
18	17	prid1 4241	. . . . . . . . . . 11 ⊢ ∅ ∈ {∅, {∅}}
19		df2o2 7461	. . . . . . . . . . 11 ⊢ 2𝑜 = {∅, {∅}}
20	18, 19	eleqtrri 2687	. . . . . . . . . 10 ⊢ ∅ ∈ 2𝑜
21	16, 20	keepel 4105	. . . . . . . . 9 ⊢ if(𝑧 ∈ 𝑦, 1𝑜, ∅) ∈ 2𝑜
22	21	rgenw 2908	. . . . . . . 8 ⊢ ∀𝑧 ∈ 𝐴 if(𝑧 ∈ 𝑦, 1𝑜, ∅) ∈ 2𝑜
23		eqid 2610	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅)) = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅))
24	23	fmpt 6289	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐴 if(𝑧 ∈ 𝑦, 1𝑜, ∅) ∈ 2𝑜 ↔ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅)):𝐴⟶2𝑜)
25	22, 24	mpbi 219	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅)):𝐴⟶2𝑜
26		simpr 476	. . . . . . . 8 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅))) → 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅)))
27	26	feq1d 5943	. . . . . . 7 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅))) → (𝑥:𝐴⟶2𝑜 ↔ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅)):𝐴⟶2𝑜))
28	25, 27	mpbiri 247	. . . . . 6 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅))) → 𝑥:𝐴⟶2𝑜)
29	26	fveq1d 6105	. . . . . . . . . . . . 13 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅))) → (𝑥‘𝑤) = ((𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅))‘𝑤))
30		elequ1 1984	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦))
31	30	ifbid 4058	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑤 → if(𝑧 ∈ 𝑦, 1𝑜, ∅) = if(𝑤 ∈ 𝑦, 1𝑜, ∅))
32	13, 17	keepel 4105	. . . . . . . . . . . . . 14 ⊢ if(𝑤 ∈ 𝑦, 1𝑜, ∅) ∈ V
33	31, 23, 32	fvmpt 6191	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ 𝐴 → ((𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅))‘𝑤) = if(𝑤 ∈ 𝑦, 1𝑜, ∅))
34	29, 33	sylan9eq 2664	. . . . . . . . . . . 12 ⊢ (((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅))) ∧ 𝑤 ∈ 𝐴) → (𝑥‘𝑤) = if(𝑤 ∈ 𝑦, 1𝑜, ∅))
35	34	eqeq1d 2612	. . . . . . . . . . 11 ⊢ (((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅))) ∧ 𝑤 ∈ 𝐴) → ((𝑥‘𝑤) = 1𝑜 ↔ if(𝑤 ∈ 𝑦, 1𝑜, ∅) = 1𝑜))
36		iftrue 4042	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝑦 → if(𝑤 ∈ 𝑦, 1𝑜, ∅) = 1𝑜)
37		noel 3878	. . . . . . . . . . . . . 14 ⊢ ¬ ∅ ∈ ∅
38		iffalse 4045	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑤 ∈ 𝑦 → if(𝑤 ∈ 𝑦, 1𝑜, ∅) = ∅)
39	38	eqeq1d 2612	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑤 ∈ 𝑦 → (if(𝑤 ∈ 𝑦, 1𝑜, ∅) = 1𝑜 ↔ ∅ = 1𝑜))
40		0lt1o 7471	. . . . . . . . . . . . . . . 16 ⊢ ∅ ∈ 1𝑜
41		eleq2 2677	. . . . . . . . . . . . . . . 16 ⊢ (∅ = 1𝑜 → (∅ ∈ ∅ ↔ ∅ ∈ 1𝑜))
42	40, 41	mpbiri 247	. . . . . . . . . . . . . . 15 ⊢ (∅ = 1𝑜 → ∅ ∈ ∅)
43	39, 42	syl6bi 242	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑤 ∈ 𝑦 → (if(𝑤 ∈ 𝑦, 1𝑜, ∅) = 1𝑜 → ∅ ∈ ∅))
44	37, 43	mtoi 189	. . . . . . . . . . . . 13 ⊢ (¬ 𝑤 ∈ 𝑦 → ¬ if(𝑤 ∈ 𝑦, 1𝑜, ∅) = 1𝑜)
45	44	con4i 112	. . . . . . . . . . . 12 ⊢ (if(𝑤 ∈ 𝑦, 1𝑜, ∅) = 1𝑜 → 𝑤 ∈ 𝑦)
46	36, 45	impbii 198	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝑦 ↔ if(𝑤 ∈ 𝑦, 1𝑜, ∅) = 1𝑜)
47	35, 46	syl6rbbr 278	. . . . . . . . . 10 ⊢ (((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅))) ∧ 𝑤 ∈ 𝐴) → (𝑤 ∈ 𝑦 ↔ (𝑥‘𝑤) = 1𝑜))
48		fvex 6113	. . . . . . . . . . 11 ⊢ (𝑥‘𝑤) ∈ V
49	48	elsn 4140	. . . . . . . . . 10 ⊢ ((𝑥‘𝑤) ∈ {1𝑜} ↔ (𝑥‘𝑤) = 1𝑜)
50	47, 49	syl6bbr 277	. . . . . . . . 9 ⊢ (((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅))) ∧ 𝑤 ∈ 𝐴) → (𝑤 ∈ 𝑦 ↔ (𝑥‘𝑤) ∈ {1𝑜}))
51	50	pm5.32da 671	. . . . . . . 8 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅))) → ((𝑤 ∈ 𝐴 ∧ 𝑤 ∈ 𝑦) ↔ (𝑤 ∈ 𝐴 ∧ (𝑥‘𝑤) ∈ {1𝑜})))
52		ssel 3562	. . . . . . . . . 10 ⊢ (𝑦 ⊆ 𝐴 → (𝑤 ∈ 𝑦 → 𝑤 ∈ 𝐴))
53	52	adantr 480	. . . . . . . . 9 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅))) → (𝑤 ∈ 𝑦 → 𝑤 ∈ 𝐴))
54	53	pm4.71rd 665	. . . . . . . 8 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅))) → (𝑤 ∈ 𝑦 ↔ (𝑤 ∈ 𝐴 ∧ 𝑤 ∈ 𝑦)))
55		ffn 5958	. . . . . . . . 9 ⊢ (𝑥:𝐴⟶2𝑜 → 𝑥 Fn 𝐴)
56		elpreima 6245	. . . . . . . . 9 ⊢ (𝑥 Fn 𝐴 → (𝑤 ∈ (◡𝑥 “ {1𝑜}) ↔ (𝑤 ∈ 𝐴 ∧ (𝑥‘𝑤) ∈ {1𝑜})))
57	28, 55, 56	3syl 18	. . . . . . . 8 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅))) → (𝑤 ∈ (◡𝑥 “ {1𝑜}) ↔ (𝑤 ∈ 𝐴 ∧ (𝑥‘𝑤) ∈ {1𝑜})))
58	51, 54, 57	3bitr4d 299	. . . . . . 7 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅))) → (𝑤 ∈ 𝑦 ↔ 𝑤 ∈ (◡𝑥 “ {1𝑜})))
59	58	eqrdv 2608	. . . . . 6 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅))) → 𝑦 = (◡𝑥 “ {1𝑜}))
60	28, 59	jca 553	. . . . 5 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅))) → (𝑥:𝐴⟶2𝑜 ∧ 𝑦 = (◡𝑥 “ {1𝑜})))
61		simpr 476	. . . . . . 7 ⊢ ((𝑥:𝐴⟶2𝑜 ∧ 𝑦 = (◡𝑥 “ {1𝑜})) → 𝑦 = (◡𝑥 “ {1𝑜}))
62		cnvimass 5404	. . . . . . . 8 ⊢ (◡𝑥 “ {1𝑜}) ⊆ dom 𝑥
63		fdm 5964	. . . . . . . . 9 ⊢ (𝑥:𝐴⟶2𝑜 → dom 𝑥 = 𝐴)
64	63	adantr 480	. . . . . . . 8 ⊢ ((𝑥:𝐴⟶2𝑜 ∧ 𝑦 = (◡𝑥 “ {1𝑜})) → dom 𝑥 = 𝐴)
65	62, 64	syl5sseq 3616	. . . . . . 7 ⊢ ((𝑥:𝐴⟶2𝑜 ∧ 𝑦 = (◡𝑥 “ {1𝑜})) → (◡𝑥 “ {1𝑜}) ⊆ 𝐴)
66	61, 65	eqsstrd 3602	. . . . . 6 ⊢ ((𝑥:𝐴⟶2𝑜 ∧ 𝑦 = (◡𝑥 “ {1𝑜})) → 𝑦 ⊆ 𝐴)
67		simplr 788	. . . . . . . . . . . . . 14 ⊢ (((𝑥:𝐴⟶2𝑜 ∧ 𝑦 = (◡𝑥 “ {1𝑜})) ∧ 𝑤 ∈ 𝐴) → 𝑦 = (◡𝑥 “ {1𝑜}))
68	67	eleq2d 2673	. . . . . . . . . . . . 13 ⊢ (((𝑥:𝐴⟶2𝑜 ∧ 𝑦 = (◡𝑥 “ {1𝑜})) ∧ 𝑤 ∈ 𝐴) → (𝑤 ∈ 𝑦 ↔ 𝑤 ∈ (◡𝑥 “ {1𝑜})))
69	55	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑥:𝐴⟶2𝑜 ∧ 𝑦 = (◡𝑥 “ {1𝑜})) → 𝑥 Fn 𝐴)
70		fnbrfvb 6146	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 Fn 𝐴 ∧ 𝑤 ∈ 𝐴) → ((𝑥‘𝑤) = 1𝑜 ↔ 𝑤𝑥1𝑜))
71	69, 70	sylan 487	. . . . . . . . . . . . . 14 ⊢ (((𝑥:𝐴⟶2𝑜 ∧ 𝑦 = (◡𝑥 “ {1𝑜})) ∧ 𝑤 ∈ 𝐴) → ((𝑥‘𝑤) = 1𝑜 ↔ 𝑤𝑥1𝑜))
72		vex 3176	. . . . . . . . . . . . . . . 16 ⊢ 𝑤 ∈ V
73	72	eliniseg 5413	. . . . . . . . . . . . . . 15 ⊢ (1𝑜 ∈ On → (𝑤 ∈ (◡𝑥 “ {1𝑜}) ↔ 𝑤𝑥1𝑜))
74	12, 73	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ (◡𝑥 “ {1𝑜}) ↔ 𝑤𝑥1𝑜)
75	71, 74	syl6bbr 277	. . . . . . . . . . . . 13 ⊢ (((𝑥:𝐴⟶2𝑜 ∧ 𝑦 = (◡𝑥 “ {1𝑜})) ∧ 𝑤 ∈ 𝐴) → ((𝑥‘𝑤) = 1𝑜 ↔ 𝑤 ∈ (◡𝑥 “ {1𝑜})))
76	68, 75	bitr4d 270	. . . . . . . . . . . 12 ⊢ (((𝑥:𝐴⟶2𝑜 ∧ 𝑦 = (◡𝑥 “ {1𝑜})) ∧ 𝑤 ∈ 𝐴) → (𝑤 ∈ 𝑦 ↔ (𝑥‘𝑤) = 1𝑜))
77	76	biimpa 500	. . . . . . . . . . 11 ⊢ ((((𝑥:𝐴⟶2𝑜 ∧ 𝑦 = (◡𝑥 “ {1𝑜})) ∧ 𝑤 ∈ 𝐴) ∧ 𝑤 ∈ 𝑦) → (𝑥‘𝑤) = 1𝑜)
78	36	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝑥:𝐴⟶2𝑜 ∧ 𝑦 = (◡𝑥 “ {1𝑜})) ∧ 𝑤 ∈ 𝐴) ∧ 𝑤 ∈ 𝑦) → if(𝑤 ∈ 𝑦, 1𝑜, ∅) = 1𝑜)
79	77, 78	eqtr4d 2647	. . . . . . . . . 10 ⊢ ((((𝑥:𝐴⟶2𝑜 ∧ 𝑦 = (◡𝑥 “ {1𝑜})) ∧ 𝑤 ∈ 𝐴) ∧ 𝑤 ∈ 𝑦) → (𝑥‘𝑤) = if(𝑤 ∈ 𝑦, 1𝑜, ∅))
80		ffvelrn 6265	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥:𝐴⟶2𝑜 ∧ 𝑤 ∈ 𝐴) → (𝑥‘𝑤) ∈ 2𝑜)
81	80	adantlr 747	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥:𝐴⟶2𝑜 ∧ 𝑦 = (◡𝑥 “ {1𝑜})) ∧ 𝑤 ∈ 𝐴) → (𝑥‘𝑤) ∈ 2𝑜)
82		df2o3 7460	. . . . . . . . . . . . . . . . 17 ⊢ 2𝑜 = {∅, 1𝑜}
83	81, 82	syl6eleq 2698	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥:𝐴⟶2𝑜 ∧ 𝑦 = (◡𝑥 “ {1𝑜})) ∧ 𝑤 ∈ 𝐴) → (𝑥‘𝑤) ∈ {∅, 1𝑜})
84	48	elpr 4146	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥‘𝑤) ∈ {∅, 1𝑜} ↔ ((𝑥‘𝑤) = ∅ ∨ (𝑥‘𝑤) = 1𝑜))
85	83, 84	sylib 207	. . . . . . . . . . . . . . 15 ⊢ (((𝑥:𝐴⟶2𝑜 ∧ 𝑦 = (◡𝑥 “ {1𝑜})) ∧ 𝑤 ∈ 𝐴) → ((𝑥‘𝑤) = ∅ ∨ (𝑥‘𝑤) = 1𝑜))
86	85	ord 391	. . . . . . . . . . . . . 14 ⊢ (((𝑥:𝐴⟶2𝑜 ∧ 𝑦 = (◡𝑥 “ {1𝑜})) ∧ 𝑤 ∈ 𝐴) → (¬ (𝑥‘𝑤) = ∅ → (𝑥‘𝑤) = 1𝑜))
87	86, 76	sylibrd 248	. . . . . . . . . . . . 13 ⊢ (((𝑥:𝐴⟶2𝑜 ∧ 𝑦 = (◡𝑥 “ {1𝑜})) ∧ 𝑤 ∈ 𝐴) → (¬ (𝑥‘𝑤) = ∅ → 𝑤 ∈ 𝑦))
88	87	con1d 138	. . . . . . . . . . . 12 ⊢ (((𝑥:𝐴⟶2𝑜 ∧ 𝑦 = (◡𝑥 “ {1𝑜})) ∧ 𝑤 ∈ 𝐴) → (¬ 𝑤 ∈ 𝑦 → (𝑥‘𝑤) = ∅))
89	88	imp 444	. . . . . . . . . . 11 ⊢ ((((𝑥:𝐴⟶2𝑜 ∧ 𝑦 = (◡𝑥 “ {1𝑜})) ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 ∈ 𝑦) → (𝑥‘𝑤) = ∅)
90	38	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝑥:𝐴⟶2𝑜 ∧ 𝑦 = (◡𝑥 “ {1𝑜})) ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 ∈ 𝑦) → if(𝑤 ∈ 𝑦, 1𝑜, ∅) = ∅)
91	89, 90	eqtr4d 2647	. . . . . . . . . 10 ⊢ ((((𝑥:𝐴⟶2𝑜 ∧ 𝑦 = (◡𝑥 “ {1𝑜})) ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 ∈ 𝑦) → (𝑥‘𝑤) = if(𝑤 ∈ 𝑦, 1𝑜, ∅))
92	79, 91	pm2.61dan 828	. . . . . . . . 9 ⊢ (((𝑥:𝐴⟶2𝑜 ∧ 𝑦 = (◡𝑥 “ {1𝑜})) ∧ 𝑤 ∈ 𝐴) → (𝑥‘𝑤) = if(𝑤 ∈ 𝑦, 1𝑜, ∅))
93	33	adantl 481	. . . . . . . . 9 ⊢ (((𝑥:𝐴⟶2𝑜 ∧ 𝑦 = (◡𝑥 “ {1𝑜})) ∧ 𝑤 ∈ 𝐴) → ((𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅))‘𝑤) = if(𝑤 ∈ 𝑦, 1𝑜, ∅))
94	92, 93	eqtr4d 2647	. . . . . . . 8 ⊢ (((𝑥:𝐴⟶2𝑜 ∧ 𝑦 = (◡𝑥 “ {1𝑜})) ∧ 𝑤 ∈ 𝐴) → (𝑥‘𝑤) = ((𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅))‘𝑤))
95	94	ralrimiva 2949	. . . . . . 7 ⊢ ((𝑥:𝐴⟶2𝑜 ∧ 𝑦 = (◡𝑥 “ {1𝑜})) → ∀𝑤 ∈ 𝐴 (𝑥‘𝑤) = ((𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅))‘𝑤))
96		ffn 5958	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅)):𝐴⟶2𝑜 → (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅)) Fn 𝐴)
97	25, 96	ax-mp 5	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅)) Fn 𝐴
98		eqfnfv 6219	. . . . . . . 8 ⊢ ((𝑥 Fn 𝐴 ∧ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅)) Fn 𝐴) → (𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅)) ↔ ∀𝑤 ∈ 𝐴 (𝑥‘𝑤) = ((𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅))‘𝑤)))
99	69, 97, 98	sylancl 693	. . . . . . 7 ⊢ ((𝑥:𝐴⟶2𝑜 ∧ 𝑦 = (◡𝑥 “ {1𝑜})) → (𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅)) ↔ ∀𝑤 ∈ 𝐴 (𝑥‘𝑤) = ((𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅))‘𝑤)))
100	95, 99	mpbird 246	. . . . . 6 ⊢ ((𝑥:𝐴⟶2𝑜 ∧ 𝑦 = (◡𝑥 “ {1𝑜})) → 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅)))
101	66, 100	jca 553	. . . . 5 ⊢ ((𝑥:𝐴⟶2𝑜 ∧ 𝑦 = (◡𝑥 “ {1𝑜})) → (𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅))))
102	60, 101	impbii 198	. . . 4 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅))) ↔ (𝑥:𝐴⟶2𝑜 ∧ 𝑦 = (◡𝑥 “ {1𝑜})))
103	11, 102	syl6bbr 277	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ∧ 𝑦 = (◡𝑥 “ {1𝑜})) ↔ (𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅)))))
104		selpw 4115	. . . 4 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴)
105	104	anbi1i 727	. . 3 ⊢ ((𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅))) ↔ (𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅))))
106	103, 105	syl6bbr 277	. 2 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (2𝑜 ↑𝑚 𝐴) ∧ 𝑦 = (◡𝑥 “ {1𝑜})) ↔ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅)))))
107	1, 5, 7, 106	f1ocnvd 6782	1 ⊢ (𝐴 ∈ 𝑉 → (𝐹:(2𝑜 ↑𝑚 𝐴)–1-1-onto→𝒫 𝐴 ∧ ◡𝐹 = (𝑦 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1𝑜, ∅)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ifcif 4036  𝒫 cpw 4108  {csn 4125  {cpr 4127   class class class wbr 4583   ↦ cmpt 4643  ^◡ccnv 5037  dom cdm 5038   “ cima 5041  Oncon0 5640  suc csuc 5642   Fn wfn 5799  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  1_𝑜c1o 7440  2_𝑜c2o 7441   ↑_𝑚 cmap 7744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1o 7447  df-2o 7448  df-map 7746

This theorem is referenced by:  pw2f1o2  36623