Theorem pw2f1ocnv 43982

Description: Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 9081, 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 3454	. . . 4 ⊢ 𝑥 ∈ V
3	2	cnvex 7920	. . 3 ⊢ ◡𝑥 ∈ V
4		imaexg 7908	. . 3 ⊢ (◡𝑥 ∈ V → (◡𝑥 “ {1o}) ∈ V)
5	3, 4	mp1i 14	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (2o ↑m 𝐴)) → (◡𝑥 “ {1o}) ∈ V)
6		mptexg 7215	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅)) ∈ V)
7	6	adantr 486	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝐴) → (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅)) ∈ V)
8		2on 8468	. . . . . 6 ⊢ 2o ∈ On
9		elmapg 8837	. . . . . 6 ⊢ ((2o ∈ On ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ (2o ↑m 𝐴) ↔ 𝑥:𝐴⟶2o))
10	8, 9	mpan 703	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (2o ↑m 𝐴) ↔ 𝑥:𝐴⟶2o))
11	10	anbi1d 643	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (2o ↑m 𝐴) ∧ 𝑦 = (◡𝑥 “ {1o})) ↔ (𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o}))))
12		1oex 8464	. . . . . . . . . . . 12 ⊢ 1o ∈ V
13	12	sucid 6436	. . . . . . . . . . 11 ⊢ 1o ∈ suc 1o
14		df-2o 8455	. . . . . . . . . . 11 ⊢ 2o = suc 1o
15	13, 14	eleqtrri 2859	. . . . . . . . . 10 ⊢ 1o ∈ 2o
16		0ex 5260	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
17	16	prid1 4722	. . . . . . . . . . 11 ⊢ ∅ ∈ {∅, {∅}}
18		df2o2 8463	. . . . . . . . . . 11 ⊢ 2o = {∅, {∅}}
19	17, 18	eleqtrri 2859	. . . . . . . . . 10 ⊢ ∅ ∈ 2o
20	15, 19	ifcli 4529	. . . . . . . . 9 ⊢ if(𝑧 ∈ 𝑦, 1o, ∅) ∈ 2o
21	20	rgenw 3080	. . . . . . . 8 ⊢ ∀𝑧 ∈ 𝐴 if(𝑧 ∈ 𝑦, 1o, ∅) ∈ 2o
22		eqid 2760	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅)) = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))
23	22	fmpt 7098	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐴 if(𝑧 ∈ 𝑦, 1o, ∅) ∈ 2o ↔ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅)):𝐴⟶2o)
24	21, 23	mpbi 233	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅)):𝐴⟶2o
25		simpr 490	. . . . . . . 8 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))) → 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅)))
26	25	feq1d 6679	. . . . . . 7 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))) → (𝑥:𝐴⟶2o ↔ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅)):𝐴⟶2o))
27	24, 26	mpbiri 261	. . . . . 6 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))) → 𝑥:𝐴⟶2o)
28		iftrue 4487	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝑦 → if(𝑤 ∈ 𝑦, 1o, ∅) = 1o)
29		noel 4283	. . . . . . . . . . . . . 14 ⊢ ¬ ∅ ∈ ∅
30		iffalse 4490	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑤 ∈ 𝑦 → if(𝑤 ∈ 𝑦, 1o, ∅) = ∅)
31	30	eqeq1d 2762	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑤 ∈ 𝑦 → (if(𝑤 ∈ 𝑦, 1o, ∅) = 1o ↔ ∅ = 1o))
32		0lt1o 8490	. . . . . . . . . . . . . . . 16 ⊢ ∅ ∈ 1o
33		eleq2 2849	. . . . . . . . . . . . . . . 16 ⊢ (∅ = 1o → (∅ ∈ ∅ ↔ ∅ ∈ 1o))
34	32, 33	mpbiri 261	. . . . . . . . . . . . . . 15 ⊢ (∅ = 1o → ∅ ∈ ∅)
35	31, 34	biimtrdi 256	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑤 ∈ 𝑦 → (if(𝑤 ∈ 𝑦, 1o, ∅) = 1o → ∅ ∈ ∅))
36	29, 35	mtoi 202	. . . . . . . . . . . . 13 ⊢ (¬ 𝑤 ∈ 𝑦 → ¬ if(𝑤 ∈ 𝑦, 1o, ∅) = 1o)
37	36	con4i 115	. . . . . . . . . . . 12 ⊢ (if(𝑤 ∈ 𝑦, 1o, ∅) = 1o → 𝑤 ∈ 𝑦)
38	28, 37	impbii 212	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝑦 ↔ if(𝑤 ∈ 𝑦, 1o, ∅) = 1o)
39	25	fveq1d 6875	. . . . . . . . . . . . 13 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))) → (𝑥‘𝑤) = ((𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))‘𝑤))
40		elequ1 2152	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦))
41	40	ifbid 4505	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑤 → if(𝑧 ∈ 𝑦, 1o, ∅) = if(𝑤 ∈ 𝑦, 1o, ∅))
42	12, 16	ifcli 4529	. . . . . . . . . . . . . 14 ⊢ if(𝑤 ∈ 𝑦, 1o, ∅) ∈ V
43	41, 22, 42	fvmpt 6981	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ 𝐴 → ((𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))‘𝑤) = if(𝑤 ∈ 𝑦, 1o, ∅))
44	39, 43	sylan9eq 2815	. . . . . . . . . . . 12 ⊢ (((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))) ∧ 𝑤 ∈ 𝐴) → (𝑥‘𝑤) = if(𝑤 ∈ 𝑦, 1o, ∅))
45	44	eqeq1d 2762	. . . . . . . . . . 11 ⊢ (((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))) ∧ 𝑤 ∈ 𝐴) → ((𝑥‘𝑤) = 1o ↔ if(𝑤 ∈ 𝑦, 1o, ∅) = 1o))
46	38, 45	bitr4id 293	. . . . . . . . . 10 ⊢ (((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))) ∧ 𝑤 ∈ 𝐴) → (𝑤 ∈ 𝑦 ↔ (𝑥‘𝑤) = 1o))
47		fvex 6886	. . . . . . . . . . 11 ⊢ (𝑥‘𝑤) ∈ V
48	47	elsn 4598	. . . . . . . . . 10 ⊢ ((𝑥‘𝑤) ∈ {1o} ↔ (𝑥‘𝑤) = 1o)
49	46, 48	bitr4di 292	. . . . . . . . 9 ⊢ (((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))) ∧ 𝑤 ∈ 𝐴) → (𝑤 ∈ 𝑦 ↔ (𝑥‘𝑤) ∈ {1o}))
50	49	pm5.32da 590	. . . . . . . 8 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))) → ((𝑤 ∈ 𝐴 ∧ 𝑤 ∈ 𝑦) ↔ (𝑤 ∈ 𝐴 ∧ (𝑥‘𝑤) ∈ {1o})))
51		ssel 3924	. . . . . . . . . 10 ⊢ (𝑦 ⊆ 𝐴 → (𝑤 ∈ 𝑦 → 𝑤 ∈ 𝐴))
52	51	adantr 486	. . . . . . . . 9 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))) → (𝑤 ∈ 𝑦 → 𝑤 ∈ 𝐴))
53	52	pm4.71rd 572	. . . . . . . 8 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))) → (𝑤 ∈ 𝑦 ↔ (𝑤 ∈ 𝐴 ∧ 𝑤 ∈ 𝑦)))
54		ffn 6697	. . . . . . . . 9 ⊢ (𝑥:𝐴⟶2o → 𝑥 Fn 𝐴)
55		elpreima 7045	. . . . . . . . 9 ⊢ (𝑥 Fn 𝐴 → (𝑤 ∈ (◡𝑥 “ {1o}) ↔ (𝑤 ∈ 𝐴 ∧ (𝑥‘𝑤) ∈ {1o})))
56	27, 54, 55	3syl 19	. . . . . . . 8 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))) → (𝑤 ∈ (◡𝑥 “ {1o}) ↔ (𝑤 ∈ 𝐴 ∧ (𝑥‘𝑤) ∈ {1o})))
57	50, 53, 56	3bitr4d 314	. . . . . . 7 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))) → (𝑤 ∈ 𝑦 ↔ 𝑤 ∈ (◡𝑥 “ {1o})))
58	57	eqrdv 2758	. . . . . 6 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))) → 𝑦 = (◡𝑥 “ {1o}))
59	27, 58	jca 521	. . . . 5 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))) → (𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})))
60		simpr 490	. . . . . . 7 ⊢ ((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) → 𝑦 = (◡𝑥 “ {1o}))
61		cnvimass 6072	. . . . . . . 8 ⊢ (◡𝑥 “ {1o}) ⊆ dom 𝑥
62		fdm 6707	. . . . . . . . 9 ⊢ (𝑥:𝐴⟶2o → dom 𝑥 = 𝐴)
63	62	adantr 486	. . . . . . . 8 ⊢ ((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) → dom 𝑥 = 𝐴)
64	61, 63	sseqtrid 3972	. . . . . . 7 ⊢ ((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) → (◡𝑥 “ {1o}) ⊆ 𝐴)
65	60, 64	eqsstrd 3964	. . . . . 6 ⊢ ((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) → 𝑦 ⊆ 𝐴)
66		simplr 781	. . . . . . . . . . . . . 14 ⊢ (((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) → 𝑦 = (◡𝑥 “ {1o}))
67	66	eleq2d 2846	. . . . . . . . . . . . 13 ⊢ (((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) → (𝑤 ∈ 𝑦 ↔ 𝑤 ∈ (◡𝑥 “ {1o})))
68	54	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) → 𝑥 Fn 𝐴)
69		fnbrfvb 6923	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 Fn 𝐴 ∧ 𝑤 ∈ 𝐴) → ((𝑥‘𝑤) = 1o ↔ 𝑤𝑥1o))
70	68, 69	sylan 592	. . . . . . . . . . . . . 14 ⊢ (((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) → ((𝑥‘𝑤) = 1o ↔ 𝑤𝑥1o))
71		1on 8467	. . . . . . . . . . . . . . 15 ⊢ 1o ∈ On
72		vex 3454	. . . . . . . . . . . . . . . 16 ⊢ 𝑤 ∈ V
73	72	eliniseg 6084	. . . . . . . . . . . . . . 15 ⊢ (1o ∈ On → (𝑤 ∈ (◡𝑥 “ {1o}) ↔ 𝑤𝑥1o))
74	71, 73	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ (◡𝑥 “ {1o}) ↔ 𝑤𝑥1o)
75	70, 74	bitr4di 292	. . . . . . . . . . . . 13 ⊢ (((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) → ((𝑥‘𝑤) = 1o ↔ 𝑤 ∈ (◡𝑥 “ {1o})))
76	67, 75	bitr4d 285	. . . . . . . . . . . 12 ⊢ (((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) → (𝑤 ∈ 𝑦 ↔ (𝑥‘𝑤) = 1o))
77	76	biimpa 482	. . . . . . . . . . 11 ⊢ ((((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) ∧ 𝑤 ∈ 𝑦) → (𝑥‘𝑤) = 1o)
78	28	adantl 487	. . . . . . . . . . 11 ⊢ ((((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) ∧ 𝑤 ∈ 𝑦) → if(𝑤 ∈ 𝑦, 1o, ∅) = 1o)
79	77, 78	eqtr4d 2798	. . . . . . . . . 10 ⊢ ((((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) ∧ 𝑤 ∈ 𝑦) → (𝑥‘𝑤) = if(𝑤 ∈ 𝑦, 1o, ∅))
80		ffvelcdm 7069	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥:𝐴⟶2o ∧ 𝑤 ∈ 𝐴) → (𝑥‘𝑤) ∈ 2o)
81	80	adantlr 728	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) → (𝑥‘𝑤) ∈ 2o)
82		df2o3 8462	. . . . . . . . . . . . . . . . 17 ⊢ 2o = {∅, 1o}
83	81, 82	eleqtrdi 2870	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) → (𝑥‘𝑤) ∈ {∅, 1o})
84	47	elpr 4608	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥‘𝑤) ∈ {∅, 1o} ↔ ((𝑥‘𝑤) = ∅ ∨ (𝑥‘𝑤) = 1o))
85	83, 84	sylib 221	. . . . . . . . . . . . . . 15 ⊢ (((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) → ((𝑥‘𝑤) = ∅ ∨ (𝑥‘𝑤) = 1o))
86	85	ord 878	. . . . . . . . . . . . . 14 ⊢ (((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) → (¬ (𝑥‘𝑤) = ∅ → (𝑥‘𝑤) = 1o))
87	86, 76	sylibrd 262	. . . . . . . . . . . . 13 ⊢ (((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) → (¬ (𝑥‘𝑤) = ∅ → 𝑤 ∈ 𝑦))
88	87	con1d 146	. . . . . . . . . . . 12 ⊢ (((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) → (¬ 𝑤 ∈ 𝑦 → (𝑥‘𝑤) = ∅))
89	88	imp 412	. . . . . . . . . . 11 ⊢ ((((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 ∈ 𝑦) → (𝑥‘𝑤) = ∅)
90	30	adantl 487	. . . . . . . . . . 11 ⊢ ((((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 ∈ 𝑦) → if(𝑤 ∈ 𝑦, 1o, ∅) = ∅)
91	89, 90	eqtr4d 2798	. . . . . . . . . 10 ⊢ ((((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 ∈ 𝑦) → (𝑥‘𝑤) = if(𝑤 ∈ 𝑦, 1o, ∅))
92	79, 91	pm2.61dan 825	. . . . . . . . 9 ⊢ (((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) → (𝑥‘𝑤) = if(𝑤 ∈ 𝑦, 1o, ∅))
93	43	adantl 487	. . . . . . . . 9 ⊢ (((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) → ((𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))‘𝑤) = if(𝑤 ∈ 𝑦, 1o, ∅))
94	92, 93	eqtr4d 2798	. . . . . . . 8 ⊢ (((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) → (𝑥‘𝑤) = ((𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))‘𝑤))
95	94	ralrimiva 3154	. . . . . . 7 ⊢ ((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) → ∀𝑤 ∈ 𝐴 (𝑥‘𝑤) = ((𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))‘𝑤))
96		ffn 6697	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅)):𝐴⟶2o → (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅)) Fn 𝐴)
97	24, 96	ax-mp 5	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅)) Fn 𝐴
98		eqfnfv 7017	. . . . . . . 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	68, 97, 98	sylancl 598	. . . . . . 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 260	. . . . . 6 ⊢ ((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) → 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅)))
101	65, 100	jca 521	. . . . 5 ⊢ ((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) → (𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))))
102	59, 101	impbii 212	. . . 4 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))) ↔ (𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})))
103	11, 102	bitr4di 292	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (2o ↑m 𝐴) ∧ 𝑦 = (◡𝑥 “ {1o})) ↔ (𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅)))))
104		velpw 4561	. . . 4 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴)
105	104	anbi1i 636	. . 3 ⊢ ((𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))) ↔ (𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))))
106	103, 105	bitr4di 292	. 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 7660	1 ⊢ (𝐴 ∈ 𝑉 → (𝐹:(2o ↑m 𝐴)–1-1-onto→𝒫 𝐴 ∧ ◡𝐹 = (𝑦 ∈ 𝒫 𝐴 ↦ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅)))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∀wral 3076  Vcvv 3450   ⊆ wss 3898  ∅c0 4278  ifcif 4481  𝒫 cpw 4556  {csn 4583  {cpr 4585   class class class wbr 5102   ↦ cmpt 5185  ^◡ccnv 5646  dom cdm 5647   “ cima 5650  Oncon0 6351  suc csuc 6353   Fn wfn 6522  ⟶wf 6523  –1-1-onto→wf1o 6526  ‘cfv 6527  (class class class)co 7408  1_oc1o 8447  2_oc2o 8448   ↑_m cmap 8825

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ord 6354  df-on 6355  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1o 8454  df-2o 8455  df-map 8827

This theorem is used by:  pw2f1o2  43983