Theorem 2omap 16270

Description: Mapping between (2_o ↑_𝑚 𝐴) and decidable subsets of 𝐴. (Contributed by Jim Kingdon, 12-Nov-2025.)

Hypothesis

Ref	Expression
2omap.f	⊢ 𝐹 = (𝑠 ∈ (2o ↑𝑚 𝐴) ↦ {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o})

Assertion

Ref	Expression
2omap	⊢ (𝐴 ∈ 𝑉 → 𝐹:(2o ↑𝑚 𝐴)–1-1-onto→{𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑥})

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

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

Proof of Theorem 2omap

Dummy variables 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2omap.f	. 2 ⊢ 𝐹 = (𝑠 ∈ (2o ↑𝑚 𝐴) ↦ {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o})
2		eleq2 2273	. . . . 5 ⊢ (𝑥 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o} → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}))
3	2	dcbid 842	. . . 4 ⊢ (𝑥 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o} → (DECID 𝑦 ∈ 𝑥 ↔ DECID 𝑦 ∈ {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}))
4	3	ralbidv 2510	. . 3 ⊢ (𝑥 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o} → (∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑥 ↔ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}))
5		ssrab2 3289	. . . . 5 ⊢ {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o} ⊆ 𝐴
6		elpw2g 4219	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ({𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o} ∈ 𝒫 𝐴 ↔ {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o} ⊆ 𝐴))
7	5, 6	mpbiri 168	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o} ∈ 𝒫 𝐴)
8	7	adantr 276	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑠 ∈ (2o ↑𝑚 𝐴)) → {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o} ∈ 𝒫 𝐴)
9		2ssom 6640	. . . . . . 7 ⊢ 2o ⊆ ω
10		elmapi 6787	. . . . . . . . 9 ⊢ (𝑠 ∈ (2o ↑𝑚 𝐴) → 𝑠:𝐴⟶2o)
11	10	ad2antlr 489	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑠 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑦 ∈ 𝐴) → 𝑠:𝐴⟶2o)
12		simpr 110	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑠 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴)
13	11, 12	ffvelcdmd 5744	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑠 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑦 ∈ 𝐴) → (𝑠‘𝑦) ∈ 2o)
14	9, 13	sselid 3202	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑠 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑦 ∈ 𝐴) → (𝑠‘𝑦) ∈ ω)
15		1onn 6636	. . . . . 6 ⊢ 1o ∈ ω
16		nndceq 6615	. . . . . 6 ⊢ (((𝑠‘𝑦) ∈ ω ∧ 1o ∈ ω) → DECID (𝑠‘𝑦) = 1o)
17	14, 15, 16	sylancl 413	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑠 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑦 ∈ 𝐴) → DECID (𝑠‘𝑦) = 1o)
18		ibar 301	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → ((𝑠‘𝑦) = 1o ↔ (𝑦 ∈ 𝐴 ∧ (𝑠‘𝑦) = 1o)))
19	18	adantl 277	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑠 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑦 ∈ 𝐴) → ((𝑠‘𝑦) = 1o ↔ (𝑦 ∈ 𝐴 ∧ (𝑠‘𝑦) = 1o)))
20		fveqeq2 5612	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → ((𝑠‘𝑧) = 1o ↔ (𝑠‘𝑦) = 1o))
21	20	elrab 2939	. . . . . . 7 ⊢ (𝑦 ∈ {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o} ↔ (𝑦 ∈ 𝐴 ∧ (𝑠‘𝑦) = 1o))
22	19, 21	bitr4di 198	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑠 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑦 ∈ 𝐴) → ((𝑠‘𝑦) = 1o ↔ 𝑦 ∈ {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}))
23	22	dcbid 842	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑠 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑦 ∈ 𝐴) → (DECID (𝑠‘𝑦) = 1o ↔ DECID 𝑦 ∈ {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}))
24	17, 23	mpbid 147	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑠 ∈ (2o ↑𝑚 𝐴)) ∧ 𝑦 ∈ 𝐴) → DECID 𝑦 ∈ {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o})
25	24	ralrimiva 2583	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑠 ∈ (2o ↑𝑚 𝐴)) → ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o})
26	4, 8, 25	elrabd 2941	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑠 ∈ (2o ↑𝑚 𝐴)) → {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o} ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑥})
27		eleq2 2273	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑤))
28	27	dcbid 842	. . . . 5 ⊢ (𝑥 = 𝑤 → (DECID 𝑦 ∈ 𝑥 ↔ DECID 𝑦 ∈ 𝑤))
29	28	ralbidv 2510	. . . 4 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑥 ↔ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))
30	29	elrab 2939	. . 3 ⊢ (𝑤 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑥} ↔ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))
31		1lt2o 6558	. . . . . . 7 ⊢ 1o ∈ 2o
32	31	a1i 9	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤)) ∧ 𝑢 ∈ 𝐴) → 1o ∈ 2o)
33		0lt2o 6557	. . . . . . 7 ⊢ ∅ ∈ 2o
34	33	a1i 9	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤)) ∧ 𝑢 ∈ 𝐴) → ∅ ∈ 2o)
35		elequ1 2184	. . . . . . . 8 ⊢ (𝑦 = 𝑢 → (𝑦 ∈ 𝑤 ↔ 𝑢 ∈ 𝑤))
36	35	dcbid 842	. . . . . . 7 ⊢ (𝑦 = 𝑢 → (DECID 𝑦 ∈ 𝑤 ↔ DECID 𝑢 ∈ 𝑤))
37		simplrr 536	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤)) ∧ 𝑢 ∈ 𝐴) → ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤)
38		simpr 110	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤)) ∧ 𝑢 ∈ 𝐴) → 𝑢 ∈ 𝐴)
39	36, 37, 38	rspcdva 2892	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤)) ∧ 𝑢 ∈ 𝐴) → DECID 𝑢 ∈ 𝑤)
40	32, 34, 39	ifcldcd 3620	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤)) ∧ 𝑢 ∈ 𝐴) → if(𝑢 ∈ 𝑤, 1o, ∅) ∈ 2o)
41	40	fmpttd 5763	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤)) → (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅)):𝐴⟶2o)
42		2onn 6637	. . . . . 6 ⊢ 2o ∈ ω
43	42	a1i 9	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤)) → 2o ∈ ω)
44		simpl 109	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤)) → 𝐴 ∈ 𝑉)
45	43, 44	elmapd 6779	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤)) → ((𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅)) ∈ (2o ↑𝑚 𝐴) ↔ (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅)):𝐴⟶2o))
46	41, 45	mpbird 167	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤)) → (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅)) ∈ (2o ↑𝑚 𝐴))
47	30, 46	sylan2b 287	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑤 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑥}) → (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅)) ∈ (2o ↑𝑚 𝐴))
48	30	anbi2i 457	. . 3 ⊢ ((𝑠 ∈ (2o ↑𝑚 𝐴) ∧ 𝑤 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑥}) ↔ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤)))
49		simpr 110	. . . . . . . 8 ⊢ (((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ∈ 𝑤) → 𝑧 ∈ 𝑤)
50		simplr 528	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) ∧ 𝑧 ∈ 𝐴) → 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅)))
51	50	fveq1d 5605	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) ∧ 𝑧 ∈ 𝐴) → (𝑠‘𝑧) = ((𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))‘𝑧))
52		eqid 2209	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅)) = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))
53		elequ1 2184	. . . . . . . . . . . . 13 ⊢ (𝑢 = 𝑧 → (𝑢 ∈ 𝑤 ↔ 𝑧 ∈ 𝑤))
54	53	ifbid 3604	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑧 → if(𝑢 ∈ 𝑤, 1o, ∅) = if(𝑧 ∈ 𝑤, 1o, ∅))
55		simpr 110	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
56	31	a1i 9	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) ∧ 𝑧 ∈ 𝐴) → 1o ∈ 2o)
57	33	a1i 9	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) ∧ 𝑧 ∈ 𝐴) → ∅ ∈ 2o)
58		elequ1 2184	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝑤 ↔ 𝑧 ∈ 𝑤))
59	58	dcbid 842	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → (DECID 𝑦 ∈ 𝑤 ↔ DECID 𝑧 ∈ 𝑤))
60		simprrr 540	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) → ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤)
61	60	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) ∧ 𝑧 ∈ 𝐴) → ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤)
62	59, 61, 55	rspcdva 2892	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) ∧ 𝑧 ∈ 𝐴) → DECID 𝑧 ∈ 𝑤)
63	56, 57, 62	ifcldcd 3620	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) ∧ 𝑧 ∈ 𝐴) → if(𝑧 ∈ 𝑤, 1o, ∅) ∈ 2o)
64	52, 54, 55, 63	fvmptd3 5701	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) ∧ 𝑧 ∈ 𝐴) → ((𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))‘𝑧) = if(𝑧 ∈ 𝑤, 1o, ∅))
65	51, 64	eqtrd 2242	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) ∧ 𝑧 ∈ 𝐴) → (𝑠‘𝑧) = if(𝑧 ∈ 𝑤, 1o, ∅))
66	65	adantr 276	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ∈ 𝑤) → (𝑠‘𝑧) = if(𝑧 ∈ 𝑤, 1o, ∅))
67	49	iftrued 3589	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ∈ 𝑤) → if(𝑧 ∈ 𝑤, 1o, ∅) = 1o)
68	66, 67	eqtrd 2242	. . . . . . . 8 ⊢ (((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ∈ 𝑤) → (𝑠‘𝑧) = 1o)
69	49, 68	2thd 175	. . . . . . 7 ⊢ (((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) ∧ 𝑧 ∈ 𝐴) ∧ 𝑧 ∈ 𝑤) → (𝑧 ∈ 𝑤 ↔ (𝑠‘𝑧) = 1o))
70		simpr 110	. . . . . . . 8 ⊢ (((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) ∧ 𝑧 ∈ 𝐴) ∧ ¬ 𝑧 ∈ 𝑤) → ¬ 𝑧 ∈ 𝑤)
71		1n0 6548	. . . . . . . . . 10 ⊢ 1o ≠ ∅
72	71	nesymi 2426	. . . . . . . . 9 ⊢ ¬ ∅ = 1o
73	65	adantr 276	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) ∧ 𝑧 ∈ 𝐴) ∧ ¬ 𝑧 ∈ 𝑤) → (𝑠‘𝑧) = if(𝑧 ∈ 𝑤, 1o, ∅))
74	70	iffalsed 3592	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) ∧ 𝑧 ∈ 𝐴) ∧ ¬ 𝑧 ∈ 𝑤) → if(𝑧 ∈ 𝑤, 1o, ∅) = ∅)
75	73, 74	eqtrd 2242	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) ∧ 𝑧 ∈ 𝐴) ∧ ¬ 𝑧 ∈ 𝑤) → (𝑠‘𝑧) = ∅)
76	75	eqeq1d 2218	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) ∧ 𝑧 ∈ 𝐴) ∧ ¬ 𝑧 ∈ 𝑤) → ((𝑠‘𝑧) = 1o ↔ ∅ = 1o))
77	72, 76	mtbiri 679	. . . . . . . 8 ⊢ (((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) ∧ 𝑧 ∈ 𝐴) ∧ ¬ 𝑧 ∈ 𝑤) → ¬ (𝑠‘𝑧) = 1o)
78	70, 77	2falsed 706	. . . . . . 7 ⊢ (((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) ∧ 𝑧 ∈ 𝐴) ∧ ¬ 𝑧 ∈ 𝑤) → (𝑧 ∈ 𝑤 ↔ (𝑠‘𝑧) = 1o))
79		exmiddc 840	. . . . . . . 8 ⊢ (DECID 𝑧 ∈ 𝑤 → (𝑧 ∈ 𝑤 ∨ ¬ 𝑧 ∈ 𝑤))
80	62, 79	syl 14	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) ∧ 𝑧 ∈ 𝐴) → (𝑧 ∈ 𝑤 ∨ ¬ 𝑧 ∈ 𝑤))
81	69, 78, 80	mpjaodan 802	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) ∧ 𝑧 ∈ 𝐴) → (𝑧 ∈ 𝑤 ↔ (𝑠‘𝑧) = 1o))
82	81	rabbidva 2767	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) → {𝑧 ∈ 𝐴 ∣ 𝑧 ∈ 𝑤} = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o})
83		elpwi 3638	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝒫 𝐴 → 𝑤 ⊆ 𝐴)
84		dfss1 3388	. . . . . . . . . 10 ⊢ (𝑤 ⊆ 𝐴 ↔ (𝐴 ∩ 𝑤) = 𝑤)
85	83, 84	sylib 122	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝒫 𝐴 → (𝐴 ∩ 𝑤) = 𝑤)
86		dfin5 3184	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝑤) = {𝑧 ∈ 𝐴 ∣ 𝑧 ∈ 𝑤}
87	85, 86	eqtr3di 2257	. . . . . . . 8 ⊢ (𝑤 ∈ 𝒫 𝐴 → 𝑤 = {𝑧 ∈ 𝐴 ∣ 𝑧 ∈ 𝑤})
88	87	eqeq1d 2218	. . . . . . 7 ⊢ (𝑤 ∈ 𝒫 𝐴 → (𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o} ↔ {𝑧 ∈ 𝐴 ∣ 𝑧 ∈ 𝑤} = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}))
89	88	ad2antrl 490	. . . . . 6 ⊢ ((𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤)) → (𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o} ↔ {𝑧 ∈ 𝐴 ∣ 𝑧 ∈ 𝑤} = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}))
90	89	ad2antlr 489	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) → (𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o} ↔ {𝑧 ∈ 𝐴 ∣ 𝑧 ∈ 𝑤} = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}))
91	82, 90	mpbird 167	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅))) → 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o})
92		simplrl 535	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) → 𝑠 ∈ (2o ↑𝑚 𝐴))
93	42	a1i 9	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) → 2o ∈ ω)
94		simpll 527	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) → 𝐴 ∈ 𝑉)
95	93, 94	elmapd 6779	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) → (𝑠 ∈ (2o ↑𝑚 𝐴) ↔ 𝑠:𝐴⟶2o))
96	92, 95	mpbid 147	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) → 𝑠:𝐴⟶2o)
97	96	feqmptd 5660	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) → 𝑠 = (𝑢 ∈ 𝐴 ↦ (𝑠‘𝑢)))
98		simpr 110	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) → 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o})
99	98	eleq2d 2279	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) → (𝑢 ∈ 𝑤 ↔ 𝑢 ∈ {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}))
100		fveqeq2 5612	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑢 → ((𝑠‘𝑧) = 1o ↔ (𝑠‘𝑢) = 1o))
101	100	elrab 2939	. . . . . . . . . . 11 ⊢ (𝑢 ∈ {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o} ↔ (𝑢 ∈ 𝐴 ∧ (𝑠‘𝑢) = 1o))
102	99, 101	bitrdi 196	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) → (𝑢 ∈ 𝑤 ↔ (𝑢 ∈ 𝐴 ∧ (𝑠‘𝑢) = 1o)))
103	102	baibd 927	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) ∧ 𝑢 ∈ 𝐴) → (𝑢 ∈ 𝑤 ↔ (𝑠‘𝑢) = 1o))
104	103	biimpa 296	. . . . . . . 8 ⊢ (((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) ∧ 𝑢 ∈ 𝐴) ∧ 𝑢 ∈ 𝑤) → (𝑠‘𝑢) = 1o)
105		simpr 110	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) ∧ 𝑢 ∈ 𝐴) ∧ 𝑢 ∈ 𝑤) → 𝑢 ∈ 𝑤)
106	105	iftrued 3589	. . . . . . . 8 ⊢ (((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) ∧ 𝑢 ∈ 𝐴) ∧ 𝑢 ∈ 𝑤) → if(𝑢 ∈ 𝑤, 1o, ∅) = 1o)
107	104, 106	eqtr4d 2245	. . . . . . 7 ⊢ (((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) ∧ 𝑢 ∈ 𝐴) ∧ 𝑢 ∈ 𝑤) → (𝑠‘𝑢) = if(𝑢 ∈ 𝑤, 1o, ∅))
108		simpr 110	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) ∧ 𝑢 ∈ 𝐴) ∧ ¬ 𝑢 ∈ 𝑤) → ¬ 𝑢 ∈ 𝑤)
109		simpr 110	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) ∧ 𝑢 ∈ 𝐴) → 𝑢 ∈ 𝐴)
110		simplr 528	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) ∧ 𝑢 ∈ 𝐴) → 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o})
111	110	eleq2d 2279	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) ∧ 𝑢 ∈ 𝐴) → (𝑢 ∈ 𝑤 ↔ 𝑢 ∈ {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}))
112	111, 101	bitrdi 196	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) ∧ 𝑢 ∈ 𝐴) → (𝑢 ∈ 𝑤 ↔ (𝑢 ∈ 𝐴 ∧ (𝑠‘𝑢) = 1o)))
113	109, 112	mpbirand 441	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) ∧ 𝑢 ∈ 𝐴) → (𝑢 ∈ 𝑤 ↔ (𝑠‘𝑢) = 1o))
114	113	adantr 276	. . . . . . . . . 10 ⊢ (((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) ∧ 𝑢 ∈ 𝐴) ∧ ¬ 𝑢 ∈ 𝑤) → (𝑢 ∈ 𝑤 ↔ (𝑠‘𝑢) = 1o))
115	108, 114	mtbid 676	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) ∧ 𝑢 ∈ 𝐴) ∧ ¬ 𝑢 ∈ 𝑤) → ¬ (𝑠‘𝑢) = 1o)
116	96	adantr 276	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) ∧ 𝑢 ∈ 𝐴) → 𝑠:𝐴⟶2o)
117	116, 109	ffvelcdmd 5744	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) ∧ 𝑢 ∈ 𝐴) → (𝑠‘𝑢) ∈ 2o)
118		df2o3 6546	. . . . . . . . . . . 12 ⊢ 2o = {∅, 1o}
119	117, 118	eleqtrdi 2302	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) ∧ 𝑢 ∈ 𝐴) → (𝑠‘𝑢) ∈ {∅, 1o})
120		elpri 3669	. . . . . . . . . . 11 ⊢ ((𝑠‘𝑢) ∈ {∅, 1o} → ((𝑠‘𝑢) = ∅ ∨ (𝑠‘𝑢) = 1o))
121	119, 120	syl 14	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) ∧ 𝑢 ∈ 𝐴) → ((𝑠‘𝑢) = ∅ ∨ (𝑠‘𝑢) = 1o))
122	121	adantr 276	. . . . . . . . 9 ⊢ (((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) ∧ 𝑢 ∈ 𝐴) ∧ ¬ 𝑢 ∈ 𝑤) → ((𝑠‘𝑢) = ∅ ∨ (𝑠‘𝑢) = 1o))
123	115, 122	ecased 1364	. . . . . . . 8 ⊢ (((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) ∧ 𝑢 ∈ 𝐴) ∧ ¬ 𝑢 ∈ 𝑤) → (𝑠‘𝑢) = ∅)
124	108	iffalsed 3592	. . . . . . . 8 ⊢ (((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) ∧ 𝑢 ∈ 𝐴) ∧ ¬ 𝑢 ∈ 𝑤) → if(𝑢 ∈ 𝑤, 1o, ∅) = ∅)
125	123, 124	eqtr4d 2245	. . . . . . 7 ⊢ (((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) ∧ 𝑢 ∈ 𝐴) ∧ ¬ 𝑢 ∈ 𝑤) → (𝑠‘𝑢) = if(𝑢 ∈ 𝑤, 1o, ∅))
126	60	ad2antrr 488	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) ∧ 𝑢 ∈ 𝐴) → ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤)
127	36, 126, 109	rspcdva 2892	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) ∧ 𝑢 ∈ 𝐴) → DECID 𝑢 ∈ 𝑤)
128		exmiddc 840	. . . . . . . 8 ⊢ (DECID 𝑢 ∈ 𝑤 → (𝑢 ∈ 𝑤 ∨ ¬ 𝑢 ∈ 𝑤))
129	127, 128	syl 14	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) ∧ 𝑢 ∈ 𝐴) → (𝑢 ∈ 𝑤 ∨ ¬ 𝑢 ∈ 𝑤))
130	107, 125, 129	mpjaodan 802	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) ∧ 𝑢 ∈ 𝐴) → (𝑠‘𝑢) = if(𝑢 ∈ 𝑤, 1o, ∅))
131	130	mpteq2dva 4153	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) → (𝑢 ∈ 𝐴 ↦ (𝑠‘𝑢)) = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅)))
132	97, 131	eqtrd 2242	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) ∧ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}) → 𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅)))
133	91, 132	impbida 598	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ (𝑤 ∈ 𝒫 𝐴 ∧ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑤))) → (𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅)) ↔ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}))
134	48, 133	sylan2b 287	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑠 ∈ (2o ↑𝑚 𝐴) ∧ 𝑤 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑥})) → (𝑠 = (𝑢 ∈ 𝐴 ↦ if(𝑢 ∈ 𝑤, 1o, ∅)) ↔ 𝑤 = {𝑧 ∈ 𝐴 ∣ (𝑠‘𝑧) = 1o}))
135	1, 26, 47, 134	f1o2d 6181	1 ⊢ (𝐴 ∈ 𝑉 → 𝐹:(2o ↑𝑚 𝐴)–1-1-onto→{𝑥 ∈ 𝒫 𝐴 ∣ ∀𝑦 ∈ 𝐴 DECID 𝑦 ∈ 𝑥})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 712  DECID wdc 838   = wceq 1375   ∈ wcel 2180  ∀wral 2488  {crab 2492   ∩ cin 3176   ⊆ wss 3177  ∅c0 3471  ifcif 3582  𝒫 cpw 3629  {cpr 3647   ↦ cmpt 4124  ωcom 4659  ⟶wf 5290  –1-1-onto→wf1o 5293  ‘cfv 5294  (class class class)co 5974  1_oc1o 6525  2_oc2o 6526   ↑_𝑚 cmap 6765

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657

This theorem depends on definitions:  df-bi 117  df-dc 839  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-rex 2494  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-if 3583  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-id 4361  df-iord 4434  df-on 4436  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1o 6532  df-2o 6533  df-map 6767

This theorem is referenced by:  2omapen  16271