Theorem pw2f1ocnv 43054

Description: Define a bijection between characteristic functions and subsets. EDITORIAL: extracted from pw2en 9120, 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 3483	. . . 4 ⊢ 𝑥 ∈ V
3	2	cnvex 7948	. . 3 ⊢ ◡𝑥 ∈ V
4		imaexg 7936	. . 3 ⊢ (◡𝑥 ∈ V → (◡𝑥 “ {1o}) ∈ V)
5	3, 4	mp1i 13	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ (2o ↑m 𝐴)) → (◡𝑥 “ {1o}) ∈ V)
6		mptexg 7242	. . 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 8521	. . . . . 6 ⊢ 2o ∈ On
9		elmapg 8880	. . . . . 6 ⊢ ((2o ∈ On ∧ 𝐴 ∈ 𝑉) → (𝑥 ∈ (2o ↑m 𝐴) ↔ 𝑥:𝐴⟶2o))
10	8, 9	mpan 690	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (2o ↑m 𝐴) ↔ 𝑥:𝐴⟶2o))
11	10	anbi1d 631	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (2o ↑m 𝐴) ∧ 𝑦 = (◡𝑥 “ {1o})) ↔ (𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o}))))
12		1oex 8517	. . . . . . . . . . . 12 ⊢ 1o ∈ V
13	12	sucid 6465	. . . . . . . . . . 11 ⊢ 1o ∈ suc 1o
14		df-2o 8508	. . . . . . . . . . 11 ⊢ 2o = suc 1o
15	13, 14	eleqtrri 2839	. . . . . . . . . 10 ⊢ 1o ∈ 2o
16		0ex 5306	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
17	16	prid1 4761	. . . . . . . . . . 11 ⊢ ∅ ∈ {∅, {∅}}
18		df2o2 8516	. . . . . . . . . . 11 ⊢ 2o = {∅, {∅}}
19	17, 18	eleqtrri 2839	. . . . . . . . . 10 ⊢ ∅ ∈ 2o
20	15, 19	ifcli 4572	. . . . . . . . 9 ⊢ if(𝑧 ∈ 𝑦, 1o, ∅) ∈ 2o
21	20	rgenw 3064	. . . . . . . 8 ⊢ ∀𝑧 ∈ 𝐴 if(𝑧 ∈ 𝑦, 1o, ∅) ∈ 2o
22		eqid 2736	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅)) = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))
23	22	fmpt 7129	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐴 if(𝑧 ∈ 𝑦, 1o, ∅) ∈ 2o ↔ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅)):𝐴⟶2o)
24	21, 23	mpbi 230	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅)):𝐴⟶2o
25		simpr 484	. . . . . . . 8 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))) → 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅)))
26	25	feq1d 6719	. . . . . . 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 258	. . . . . 6 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))) → 𝑥:𝐴⟶2o)
28		iftrue 4530	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝑦 → if(𝑤 ∈ 𝑦, 1o, ∅) = 1o)
29		noel 4337	. . . . . . . . . . . . . 14 ⊢ ¬ ∅ ∈ ∅
30		iffalse 4533	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑤 ∈ 𝑦 → if(𝑤 ∈ 𝑦, 1o, ∅) = ∅)
31	30	eqeq1d 2738	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑤 ∈ 𝑦 → (if(𝑤 ∈ 𝑦, 1o, ∅) = 1o ↔ ∅ = 1o))
32		0lt1o 8543	. . . . . . . . . . . . . . . 16 ⊢ ∅ ∈ 1o
33		eleq2 2829	. . . . . . . . . . . . . . . 16 ⊢ (∅ = 1o → (∅ ∈ ∅ ↔ ∅ ∈ 1o))
34	32, 33	mpbiri 258	. . . . . . . . . . . . . . 15 ⊢ (∅ = 1o → ∅ ∈ ∅)
35	31, 34	biimtrdi 253	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑤 ∈ 𝑦 → (if(𝑤 ∈ 𝑦, 1o, ∅) = 1o → ∅ ∈ ∅))
36	29, 35	mtoi 199	. . . . . . . . . . . . 13 ⊢ (¬ 𝑤 ∈ 𝑦 → ¬ if(𝑤 ∈ 𝑦, 1o, ∅) = 1o)
37	36	con4i 114	. . . . . . . . . . . 12 ⊢ (if(𝑤 ∈ 𝑦, 1o, ∅) = 1o → 𝑤 ∈ 𝑦)
38	28, 37	impbii 209	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝑦 ↔ if(𝑤 ∈ 𝑦, 1o, ∅) = 1o)
39	25	fveq1d 6907	. . . . . . . . . . . . 13 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))) → (𝑥‘𝑤) = ((𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))‘𝑤))
40		elequ1 2114	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ 𝑦 ↔ 𝑤 ∈ 𝑦))
41	40	ifbid 4548	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑤 → if(𝑧 ∈ 𝑦, 1o, ∅) = if(𝑤 ∈ 𝑦, 1o, ∅))
42	12, 16	ifcli 4572	. . . . . . . . . . . . . 14 ⊢ if(𝑤 ∈ 𝑦, 1o, ∅) ∈ V
43	41, 22, 42	fvmpt 7015	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ 𝐴 → ((𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))‘𝑤) = if(𝑤 ∈ 𝑦, 1o, ∅))
44	39, 43	sylan9eq 2796	. . . . . . . . . . . 12 ⊢ (((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))) ∧ 𝑤 ∈ 𝐴) → (𝑥‘𝑤) = if(𝑤 ∈ 𝑦, 1o, ∅))
45	44	eqeq1d 2738	. . . . . . . . . . 11 ⊢ (((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))) ∧ 𝑤 ∈ 𝐴) → ((𝑥‘𝑤) = 1o ↔ if(𝑤 ∈ 𝑦, 1o, ∅) = 1o))
46	38, 45	bitr4id 290	. . . . . . . . . 10 ⊢ (((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))) ∧ 𝑤 ∈ 𝐴) → (𝑤 ∈ 𝑦 ↔ (𝑥‘𝑤) = 1o))
47		fvex 6918	. . . . . . . . . . 11 ⊢ (𝑥‘𝑤) ∈ V
48	47	elsn 4640	. . . . . . . . . 10 ⊢ ((𝑥‘𝑤) ∈ {1o} ↔ (𝑥‘𝑤) = 1o)
49	46, 48	bitr4di 289	. . . . . . . . 9 ⊢ (((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))) ∧ 𝑤 ∈ 𝐴) → (𝑤 ∈ 𝑦 ↔ (𝑥‘𝑤) ∈ {1o}))
50	49	pm5.32da 579	. . . . . . . 8 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))) → ((𝑤 ∈ 𝐴 ∧ 𝑤 ∈ 𝑦) ↔ (𝑤 ∈ 𝐴 ∧ (𝑥‘𝑤) ∈ {1o})))
51		ssel 3976	. . . . . . . . . 10 ⊢ (𝑦 ⊆ 𝐴 → (𝑤 ∈ 𝑦 → 𝑤 ∈ 𝐴))
52	51	adantr 480	. . . . . . . . 9 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))) → (𝑤 ∈ 𝑦 → 𝑤 ∈ 𝐴))
53	52	pm4.71rd 562	. . . . . . . 8 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))) → (𝑤 ∈ 𝑦 ↔ (𝑤 ∈ 𝐴 ∧ 𝑤 ∈ 𝑦)))
54		ffn 6735	. . . . . . . . 9 ⊢ (𝑥:𝐴⟶2o → 𝑥 Fn 𝐴)
55		elpreima 7077	. . . . . . . . 9 ⊢ (𝑥 Fn 𝐴 → (𝑤 ∈ (◡𝑥 “ {1o}) ↔ (𝑤 ∈ 𝐴 ∧ (𝑥‘𝑤) ∈ {1o})))
56	27, 54, 55	3syl 18	. . . . . . . 8 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))) → (𝑤 ∈ (◡𝑥 “ {1o}) ↔ (𝑤 ∈ 𝐴 ∧ (𝑥‘𝑤) ∈ {1o})))
57	50, 53, 56	3bitr4d 311	. . . . . . 7 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))) → (𝑤 ∈ 𝑦 ↔ 𝑤 ∈ (◡𝑥 “ {1o})))
58	57	eqrdv 2734	. . . . . 6 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))) → 𝑦 = (◡𝑥 “ {1o}))
59	27, 58	jca 511	. . . . 5 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))) → (𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})))
60		simpr 484	. . . . . . 7 ⊢ ((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) → 𝑦 = (◡𝑥 “ {1o}))
61		cnvimass 6099	. . . . . . . 8 ⊢ (◡𝑥 “ {1o}) ⊆ dom 𝑥
62		fdm 6744	. . . . . . . . 9 ⊢ (𝑥:𝐴⟶2o → dom 𝑥 = 𝐴)
63	62	adantr 480	. . . . . . . 8 ⊢ ((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) → dom 𝑥 = 𝐴)
64	61, 63	sseqtrid 4025	. . . . . . 7 ⊢ ((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) → (◡𝑥 “ {1o}) ⊆ 𝐴)
65	60, 64	eqsstrd 4017	. . . . . 6 ⊢ ((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) → 𝑦 ⊆ 𝐴)
66		simplr 768	. . . . . . . . . . . . . 14 ⊢ (((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) → 𝑦 = (◡𝑥 “ {1o}))
67	66	eleq2d 2826	. . . . . . . . . . . . 13 ⊢ (((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) → (𝑤 ∈ 𝑦 ↔ 𝑤 ∈ (◡𝑥 “ {1o})))
68	54	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) → 𝑥 Fn 𝐴)
69		fnbrfvb 6958	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 Fn 𝐴 ∧ 𝑤 ∈ 𝐴) → ((𝑥‘𝑤) = 1o ↔ 𝑤𝑥1o))
70	68, 69	sylan 580	. . . . . . . . . . . . . 14 ⊢ (((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) → ((𝑥‘𝑤) = 1o ↔ 𝑤𝑥1o))
71		1on 8519	. . . . . . . . . . . . . . 15 ⊢ 1o ∈ On
72		vex 3483	. . . . . . . . . . . . . . . 16 ⊢ 𝑤 ∈ V
73	72	eliniseg 6111	. . . . . . . . . . . . . . 15 ⊢ (1o ∈ On → (𝑤 ∈ (◡𝑥 “ {1o}) ↔ 𝑤𝑥1o))
74	71, 73	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ (◡𝑥 “ {1o}) ↔ 𝑤𝑥1o)
75	70, 74	bitr4di 289	. . . . . . . . . . . . 13 ⊢ (((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) → ((𝑥‘𝑤) = 1o ↔ 𝑤 ∈ (◡𝑥 “ {1o})))
76	67, 75	bitr4d 282	. . . . . . . . . . . 12 ⊢ (((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) → (𝑤 ∈ 𝑦 ↔ (𝑥‘𝑤) = 1o))
77	76	biimpa 476	. . . . . . . . . . 11 ⊢ ((((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) ∧ 𝑤 ∈ 𝑦) → (𝑥‘𝑤) = 1o)
78	28	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) ∧ 𝑤 ∈ 𝑦) → if(𝑤 ∈ 𝑦, 1o, ∅) = 1o)
79	77, 78	eqtr4d 2779	. . . . . . . . . 10 ⊢ ((((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) ∧ 𝑤 ∈ 𝑦) → (𝑥‘𝑤) = if(𝑤 ∈ 𝑦, 1o, ∅))
80		ffvelcdm 7100	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥:𝐴⟶2o ∧ 𝑤 ∈ 𝐴) → (𝑥‘𝑤) ∈ 2o)
81	80	adantlr 715	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) → (𝑥‘𝑤) ∈ 2o)
82		df2o3 8515	. . . . . . . . . . . . . . . . 17 ⊢ 2o = {∅, 1o}
83	81, 82	eleqtrdi 2850	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) → (𝑥‘𝑤) ∈ {∅, 1o})
84	47	elpr 4649	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥‘𝑤) ∈ {∅, 1o} ↔ ((𝑥‘𝑤) = ∅ ∨ (𝑥‘𝑤) = 1o))
85	83, 84	sylib 218	. . . . . . . . . . . . . . 15 ⊢ (((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) → ((𝑥‘𝑤) = ∅ ∨ (𝑥‘𝑤) = 1o))
86	85	ord 864	. . . . . . . . . . . . . 14 ⊢ (((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) → (¬ (𝑥‘𝑤) = ∅ → (𝑥‘𝑤) = 1o))
87	86, 76	sylibrd 259	. . . . . . . . . . . . 13 ⊢ (((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) → (¬ (𝑥‘𝑤) = ∅ → 𝑤 ∈ 𝑦))
88	87	con1d 145	. . . . . . . . . . . 12 ⊢ (((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) → (¬ 𝑤 ∈ 𝑦 → (𝑥‘𝑤) = ∅))
89	88	imp 406	. . . . . . . . . . 11 ⊢ ((((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 ∈ 𝑦) → (𝑥‘𝑤) = ∅)
90	30	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 ∈ 𝑦) → if(𝑤 ∈ 𝑦, 1o, ∅) = ∅)
91	89, 90	eqtr4d 2779	. . . . . . . . . 10 ⊢ ((((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) ∧ ¬ 𝑤 ∈ 𝑦) → (𝑥‘𝑤) = if(𝑤 ∈ 𝑦, 1o, ∅))
92	79, 91	pm2.61dan 812	. . . . . . . . 9 ⊢ (((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) → (𝑥‘𝑤) = if(𝑤 ∈ 𝑦, 1o, ∅))
93	43	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 2779	. . . . . . . 8 ⊢ (((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) ∧ 𝑤 ∈ 𝐴) → (𝑥‘𝑤) = ((𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))‘𝑤))
95	94	ralrimiva 3145	. . . . . . 7 ⊢ ((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) → ∀𝑤 ∈ 𝐴 (𝑥‘𝑤) = ((𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))‘𝑤))
96		ffn 6735	. . . . . . . . 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 7050	. . . . . . . 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 586	. . . . . . 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 257	. . . . . 6 ⊢ ((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) → 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅)))
101	65, 100	jca 511	. . . . 5 ⊢ ((𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})) → (𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))))
102	59, 101	impbii 209	. . . 4 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))) ↔ (𝑥:𝐴⟶2o ∧ 𝑦 = (◡𝑥 “ {1o})))
103	11, 102	bitr4di 289	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((𝑥 ∈ (2o ↑m 𝐴) ∧ 𝑦 = (◡𝑥 “ {1o})) ↔ (𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅)))))
104		velpw 4604	. . . 4 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴)
105	104	anbi1i 624	. . 3 ⊢ ((𝑦 ∈ 𝒫 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))) ↔ (𝑦 ⊆ 𝐴 ∧ 𝑥 = (𝑧 ∈ 𝐴 ↦ if(𝑧 ∈ 𝑦, 1o, ∅))))
106	103, 105	bitr4di 289	. 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 7685	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 206   ∧ wa 395   ∨ wo 847   = wceq 1539   ∈ wcel 2107  ∀wral 3060  Vcvv 3479   ⊆ wss 3950  ∅c0 4332  ifcif 4524  𝒫 cpw 4599  {csn 4625  {cpr 4627   class class class wbr 5142   ↦ cmpt 5224  ^◡ccnv 5683  dom cdm 5684   “ cima 5687  Oncon0 6383  suc csuc 6385   Fn wfn 6555  ⟶wf 6556  –1-1-onto→wf1o 6559  ‘cfv 6560  (class class class)co 7432  1_oc1o 8500  2_oc2o 8501   ↑_m cmap 8867

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 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

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 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1o 8507  df-2o 8508  df-map 8869

This theorem is referenced by:  pw2f1o2  43055