Theorem bj-charfunbi 13846

Description: In an ambient set 𝑋, if membership in 𝐴 is stable, then it is decidable if and only if 𝐴 has a characteristic function.

This characterization can be applied to singletons when the set 𝑋 has stable equality, which is the case as soon as it has a tight apartness relation. (Contributed by BJ, 6-Aug-2024.)

Hypotheses

Ref	Expression
bj-charfunbi.ex	⊢ (𝜑 → 𝑋 ∈ 𝑉)
bj-charfunbi.st	⊢ (𝜑 → ∀𝑥 ∈ 𝑋 STAB 𝑥 ∈ 𝐴)

Assertion

Ref	Expression
bj-charfunbi	⊢ (𝜑 → (∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴 ↔ ∃𝑓 ∈ (2o ↑𝑚 𝑋)(∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅)))

Distinct variable groups:   𝐴,𝑓,𝑥   𝑓,𝑋,𝑥   𝜑,𝑓,𝑥

Allowed substitution hints:   𝑉(𝑥,𝑓)

Proof of Theorem bj-charfunbi

Dummy variables 𝑔 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1w 2231	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
2	1	dcbid 833	. . . 4 ⊢ (𝑥 = 𝑧 → (DECID 𝑥 ∈ 𝐴 ↔ DECID 𝑧 ∈ 𝐴))
3	2	cbvralvw 2700	. . 3 ⊢ (∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴 ↔ ∀𝑧 ∈ 𝑋 DECID 𝑧 ∈ 𝐴)
4		eleq1w 2231	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
5	4	ifbid 3547	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → if(𝑧 ∈ 𝐴, 1o, ∅) = if(𝑥 ∈ 𝐴, 1o, ∅))
6	5	cbvmptv 4085	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅)) = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅))
7	6	a1i 9	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝑋 DECID 𝑧 ∈ 𝐴) → (𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅)) = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅)))
8	3	biimpri 132	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ 𝑋 DECID 𝑧 ∈ 𝐴 → ∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴)
9	8	adantl 275	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝑋 DECID 𝑧 ∈ 𝐴) → ∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴)
10	7, 9	bj-charfundc 13843	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝑋 DECID 𝑧 ∈ 𝐴) → ((𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅)):𝑋⟶2o ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)((𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅))‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)((𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅))‘𝑥) = ∅)))
11	10	ex 114	. . . . . . 7 ⊢ (𝜑 → (∀𝑧 ∈ 𝑋 DECID 𝑧 ∈ 𝐴 → ((𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅)):𝑋⟶2o ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)((𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅))‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)((𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅))‘𝑥) = ∅))))
12		2on 6404	. . . . . . . . . . 11 ⊢ 2o ∈ On
13	12	a1i 9	. . . . . . . . . 10 ⊢ (𝜑 → 2o ∈ On)
14		bj-charfunbi.ex	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
15	13, 14	elmapd 6640	. . . . . . . . 9 ⊢ (𝜑 → ((𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅)) ∈ (2o ↑𝑚 𝑋) ↔ (𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅)):𝑋⟶2o))
16	15	biimprd 157	. . . . . . . 8 ⊢ (𝜑 → ((𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅)):𝑋⟶2o → (𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅)) ∈ (2o ↑𝑚 𝑋)))
17	16	adantrd 277	. . . . . . 7 ⊢ (𝜑 → (((𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅)):𝑋⟶2o ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)((𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅))‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)((𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅))‘𝑥) = ∅)) → (𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅)) ∈ (2o ↑𝑚 𝑋)))
18	11, 17	syld 45	. . . . . 6 ⊢ (𝜑 → (∀𝑧 ∈ 𝑋 DECID 𝑧 ∈ 𝐴 → (𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅)) ∈ (2o ↑𝑚 𝑋)))
19	18	imp 123	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝑋 DECID 𝑧 ∈ 𝐴) → (𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅)) ∈ (2o ↑𝑚 𝑋))
20		fveq1 5495	. . . . . . . . 9 ⊢ (𝑓 = (𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅)) → (𝑓‘𝑥) = ((𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅))‘𝑥))
21	20	eqeq1d 2179	. . . . . . . 8 ⊢ (𝑓 = (𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅)) → ((𝑓‘𝑥) = 1o ↔ ((𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅))‘𝑥) = 1o))
22	21	ralbidv 2470	. . . . . . 7 ⊢ (𝑓 = (𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅)) → (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) = 1o ↔ ∀𝑥 ∈ (𝑋 ∩ 𝐴)((𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅))‘𝑥) = 1o))
23	20	eqeq1d 2179	. . . . . . . 8 ⊢ (𝑓 = (𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅)) → ((𝑓‘𝑥) = ∅ ↔ ((𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅))‘𝑥) = ∅))
24	23	ralbidv 2470	. . . . . . 7 ⊢ (𝑓 = (𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅)) → (∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅ ↔ ∀𝑥 ∈ (𝑋 ∖ 𝐴)((𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅))‘𝑥) = ∅))
25	22, 24	anbi12d 470	. . . . . 6 ⊢ (𝑓 = (𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅)) → ((∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅) ↔ (∀𝑥 ∈ (𝑋 ∩ 𝐴)((𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅))‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)((𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅))‘𝑥) = ∅)))
26	25	adantl 275	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑧 ∈ 𝑋 DECID 𝑧 ∈ 𝐴) ∧ 𝑓 = (𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅))) → ((∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅) ↔ (∀𝑥 ∈ (𝑋 ∩ 𝐴)((𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅))‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)((𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅))‘𝑥) = ∅)))
27	10	simprd 113	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝑋 DECID 𝑧 ∈ 𝐴) → (∀𝑥 ∈ (𝑋 ∩ 𝐴)((𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅))‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)((𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅))‘𝑥) = ∅))
28	19, 26, 27	rspcedvd 2840	. . . 4 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝑋 DECID 𝑧 ∈ 𝐴) → ∃𝑓 ∈ (2o ↑𝑚 𝑋)(∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅))
29	28	ex 114	. . 3 ⊢ (𝜑 → (∀𝑧 ∈ 𝑋 DECID 𝑧 ∈ 𝐴 → ∃𝑓 ∈ (2o ↑𝑚 𝑋)(∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅)))
30	3, 29	syl5bi 151	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴 → ∃𝑓 ∈ (2o ↑𝑚 𝑋)(∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅)))
31		omex 4577	. . . . . . . . 9 ⊢ ω ∈ V
32		2ssom 6503	. . . . . . . . 9 ⊢ 2o ⊆ ω
33		mapss 6669	. . . . . . . . 9 ⊢ ((ω ∈ V ∧ 2o ⊆ ω) → (2o ↑𝑚 𝑋) ⊆ (ω ↑𝑚 𝑋))
34	31, 32, 33	mp2an 424	. . . . . . . 8 ⊢ (2o ↑𝑚 𝑋) ⊆ (ω ↑𝑚 𝑋)
35		fveq1 5495	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑥) = (𝑔‘𝑥))
36	35	eqeq1d 2179	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → ((𝑓‘𝑥) = 1o ↔ (𝑔‘𝑥) = 1o))
37	36	ralbidv 2470	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) = 1o ↔ ∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑔‘𝑥) = 1o))
38	35	eqeq1d 2179	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → ((𝑓‘𝑥) = ∅ ↔ (𝑔‘𝑥) = ∅))
39	38	ralbidv 2470	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅ ↔ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑔‘𝑥) = ∅))
40	37, 39	anbi12d 470	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → ((∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅) ↔ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑔‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑔‘𝑥) = ∅)))
41	40	cbvrexvw 2701	. . . . . . . . 9 ⊢ (∃𝑓 ∈ (2o ↑𝑚 𝑋)(∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅) ↔ ∃𝑔 ∈ (2o ↑𝑚 𝑋)(∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑔‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑔‘𝑥) = ∅))
42		fveqeq2 5505	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((𝑔‘𝑥) = 1o ↔ (𝑔‘𝑦) = 1o))
43	42	cbvralvw 2700	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑔‘𝑥) = 1o ↔ ∀𝑦 ∈ (𝑋 ∩ 𝐴)(𝑔‘𝑦) = 1o)
44		1n0 6411	. . . . . . . . . . . . . . . 16 ⊢ 1o ≠ ∅
45	44	neii 2342	. . . . . . . . . . . . . . 15 ⊢ ¬ 1o = ∅
46		eqeq1 2177	. . . . . . . . . . . . . . 15 ⊢ ((𝑔‘𝑦) = 1o → ((𝑔‘𝑦) = ∅ ↔ 1o = ∅))
47	45, 46	mtbiri 670	. . . . . . . . . . . . . 14 ⊢ ((𝑔‘𝑦) = 1o → ¬ (𝑔‘𝑦) = ∅)
48	47	neqned 2347	. . . . . . . . . . . . 13 ⊢ ((𝑔‘𝑦) = 1o → (𝑔‘𝑦) ≠ ∅)
49	48	ralimi 2533	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ (𝑋 ∩ 𝐴)(𝑔‘𝑦) = 1o → ∀𝑦 ∈ (𝑋 ∩ 𝐴)(𝑔‘𝑦) ≠ ∅)
50	43, 49	sylbi 120	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑔‘𝑥) = 1o → ∀𝑦 ∈ (𝑋 ∩ 𝐴)(𝑔‘𝑦) ≠ ∅)
51		fveqeq2 5505	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((𝑔‘𝑥) = ∅ ↔ (𝑔‘𝑦) = ∅))
52	51	cbvralvw 2700	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑔‘𝑥) = ∅ ↔ ∀𝑦 ∈ (𝑋 ∖ 𝐴)(𝑔‘𝑦) = ∅)
53	52	biimpi 119	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑔‘𝑥) = ∅ → ∀𝑦 ∈ (𝑋 ∖ 𝐴)(𝑔‘𝑦) = ∅)
54	50, 53	anim12i 336	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑔‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑔‘𝑥) = ∅) → (∀𝑦 ∈ (𝑋 ∩ 𝐴)(𝑔‘𝑦) ≠ ∅ ∧ ∀𝑦 ∈ (𝑋 ∖ 𝐴)(𝑔‘𝑦) = ∅))
55	54	reximi 2567	. . . . . . . . 9 ⊢ (∃𝑔 ∈ (2o ↑𝑚 𝑋)(∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑔‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑔‘𝑥) = ∅) → ∃𝑔 ∈ (2o ↑𝑚 𝑋)(∀𝑦 ∈ (𝑋 ∩ 𝐴)(𝑔‘𝑦) ≠ ∅ ∧ ∀𝑦 ∈ (𝑋 ∖ 𝐴)(𝑔‘𝑦) = ∅))
56	41, 55	sylbi 120	. . . . . . . 8 ⊢ (∃𝑓 ∈ (2o ↑𝑚 𝑋)(∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅) → ∃𝑔 ∈ (2o ↑𝑚 𝑋)(∀𝑦 ∈ (𝑋 ∩ 𝐴)(𝑔‘𝑦) ≠ ∅ ∧ ∀𝑦 ∈ (𝑋 ∖ 𝐴)(𝑔‘𝑦) = ∅))
57		ssrexv 3212	. . . . . . . 8 ⊢ ((2o ↑𝑚 𝑋) ⊆ (ω ↑𝑚 𝑋) → (∃𝑔 ∈ (2o ↑𝑚 𝑋)(∀𝑦 ∈ (𝑋 ∩ 𝐴)(𝑔‘𝑦) ≠ ∅ ∧ ∀𝑦 ∈ (𝑋 ∖ 𝐴)(𝑔‘𝑦) = ∅) → ∃𝑔 ∈ (ω ↑𝑚 𝑋)(∀𝑦 ∈ (𝑋 ∩ 𝐴)(𝑔‘𝑦) ≠ ∅ ∧ ∀𝑦 ∈ (𝑋 ∖ 𝐴)(𝑔‘𝑦) = ∅)))
58	34, 56, 57	mpsyl 65	. . . . . . 7 ⊢ (∃𝑓 ∈ (2o ↑𝑚 𝑋)(∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅) → ∃𝑔 ∈ (ω ↑𝑚 𝑋)(∀𝑦 ∈ (𝑋 ∩ 𝐴)(𝑔‘𝑦) ≠ ∅ ∧ ∀𝑦 ∈ (𝑋 ∖ 𝐴)(𝑔‘𝑦) = ∅))
59	58	adantl 275	. . . . . 6 ⊢ ((𝜑 ∧ ∃𝑓 ∈ (2o ↑𝑚 𝑋)(∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅)) → ∃𝑔 ∈ (ω ↑𝑚 𝑋)(∀𝑦 ∈ (𝑋 ∩ 𝐴)(𝑔‘𝑦) ≠ ∅ ∧ ∀𝑦 ∈ (𝑋 ∖ 𝐴)(𝑔‘𝑦) = ∅))
60	59	bj-charfunr 13845	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑓 ∈ (2o ↑𝑚 𝑋)(∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅)) → ∀𝑦 ∈ 𝑋 DECID ¬ 𝑦 ∈ 𝐴)
61	60	ex 114	. . . 4 ⊢ (𝜑 → (∃𝑓 ∈ (2o ↑𝑚 𝑋)(∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅) → ∀𝑦 ∈ 𝑋 DECID ¬ 𝑦 ∈ 𝐴))
62		eleq1w 2231	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
63	62	notbid 662	. . . . . 6 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝐴 ↔ ¬ 𝑦 ∈ 𝐴))
64	63	dcbid 833	. . . . 5 ⊢ (𝑥 = 𝑦 → (DECID ¬ 𝑥 ∈ 𝐴 ↔ DECID ¬ 𝑦 ∈ 𝐴))
65	64	cbvralvw 2700	. . . 4 ⊢ (∀𝑥 ∈ 𝑋 DECID ¬ 𝑥 ∈ 𝐴 ↔ ∀𝑦 ∈ 𝑋 DECID ¬ 𝑦 ∈ 𝐴)
66	61, 65	syl6ibr 161	. . 3 ⊢ (𝜑 → (∃𝑓 ∈ (2o ↑𝑚 𝑋)(∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅) → ∀𝑥 ∈ 𝑋 DECID ¬ 𝑥 ∈ 𝐴))
67		bj-charfunbi.st	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 STAB 𝑥 ∈ 𝐴)
68	67	r19.21bi 2558	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → STAB 𝑥 ∈ 𝐴)
69		stdcn 842	. . . . 5 ⊢ (STAB 𝑥 ∈ 𝐴 ↔ (DECID ¬ 𝑥 ∈ 𝐴 → DECID 𝑥 ∈ 𝐴))
70	68, 69	sylib 121	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (DECID ¬ 𝑥 ∈ 𝐴 → DECID 𝑥 ∈ 𝐴))
71	70	ralimdva 2537	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝑋 DECID ¬ 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴))
72	66, 71	syld 45	. 2 ⊢ (𝜑 → (∃𝑓 ∈ (2o ↑𝑚 𝑋)(∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅) → ∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴))
73	30, 72	impbid 128	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴 ↔ ∃𝑓 ∈ (2o ↑𝑚 𝑋)(∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  STAB wstab 825  DECID wdc 829   = wceq 1348   ∈ wcel 2141   ≠ wne 2340  ∀wral 2448  ∃wrex 2449  Vcvv 2730   ∖ cdif 3118   ∩ cin 3120   ⊆ wss 3121  ∅c0 3414  ifcif 3526   ↦ cmpt 4050  Oncon0 4348  ωcom 4574  ⟶wf 5194  ‘cfv 5198  (class class class)co 5853  1_oc1o 6388  2_oc2o 6389   ↑_𝑚 cmap 6626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1o 6395  df-2o 6396  df-map 6628

This theorem is referenced by: (None)