Theorem bj-charfunbi 16406

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 2292	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴))
2	1	dcbid 845	. . . 4 ⊢ (𝑥 = 𝑧 → (DECID 𝑥 ∈ 𝐴 ↔ DECID 𝑧 ∈ 𝐴))
3	2	cbvralvw 2771	. . 3 ⊢ (∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴 ↔ ∀𝑧 ∈ 𝑋 DECID 𝑧 ∈ 𝐴)
4		eleq1w 2292	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
5	4	ifbid 3627	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → if(𝑧 ∈ 𝐴, 1o, ∅) = if(𝑥 ∈ 𝐴, 1o, ∅))
6	5	cbvmptv 4185	. . . . . . . . . 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 133	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ 𝑋 DECID 𝑧 ∈ 𝐴 → ∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴)
9	8	adantl 277	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝑋 DECID 𝑧 ∈ 𝐴) → ∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴)
10	7, 9	bj-charfundc 16403	. . . . . . . 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 115	. . . . . . 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 6590	. . . . . . . . . . 11 ⊢ 2o ∈ On
13	12	a1i 9	. . . . . . . . . 10 ⊢ (𝜑 → 2o ∈ On)
14		bj-charfunbi.ex	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
15	13, 14	elmapd 6830	. . . . . . . . 9 ⊢ (𝜑 → ((𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅)) ∈ (2o ↑𝑚 𝑋) ↔ (𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅)):𝑋⟶2o))
16	15	biimprd 158	. . . . . . . 8 ⊢ (𝜑 → ((𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅)):𝑋⟶2o → (𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅)) ∈ (2o ↑𝑚 𝑋)))
17	16	adantrd 279	. . . . . . 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 124	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑧 ∈ 𝑋 DECID 𝑧 ∈ 𝐴) → (𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅)) ∈ (2o ↑𝑚 𝑋))
20		fveq1 5638	. . . . . . . . 9 ⊢ (𝑓 = (𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅)) → (𝑓‘𝑥) = ((𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅))‘𝑥))
21	20	eqeq1d 2240	. . . . . . . 8 ⊢ (𝑓 = (𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅)) → ((𝑓‘𝑥) = 1o ↔ ((𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅))‘𝑥) = 1o))
22	21	ralbidv 2532	. . . . . . 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 2240	. . . . . . . 8 ⊢ (𝑓 = (𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅)) → ((𝑓‘𝑥) = ∅ ↔ ((𝑧 ∈ 𝑋 ↦ if(𝑧 ∈ 𝐴, 1o, ∅))‘𝑥) = ∅))
24	23	ralbidv 2532	. . . . . . 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 473	. . . . . 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 277	. . . . 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 114	. . . . 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 2916	. . . 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 115	. . 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	biimtrid 152	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴 → ∃𝑓 ∈ (2o ↑𝑚 𝑋)(∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅)))
31		omex 4691	. . . . . . . . 9 ⊢ ω ∈ V
32		2ssom 6691	. . . . . . . . 9 ⊢ 2o ⊆ ω
33		mapss 6859	. . . . . . . . 9 ⊢ ((ω ∈ V ∧ 2o ⊆ ω) → (2o ↑𝑚 𝑋) ⊆ (ω ↑𝑚 𝑋))
34	31, 32, 33	mp2an 426	. . . . . . . 8 ⊢ (2o ↑𝑚 𝑋) ⊆ (ω ↑𝑚 𝑋)
35		fveq1 5638	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑥) = (𝑔‘𝑥))
36	35	eqeq1d 2240	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → ((𝑓‘𝑥) = 1o ↔ (𝑔‘𝑥) = 1o))
37	36	ralbidv 2532	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) = 1o ↔ ∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑔‘𝑥) = 1o))
38	35	eqeq1d 2240	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → ((𝑓‘𝑥) = ∅ ↔ (𝑔‘𝑥) = ∅))
39	38	ralbidv 2532	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅ ↔ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑔‘𝑥) = ∅))
40	37, 39	anbi12d 473	. . . . . . . . . 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 2772	. . . . . . . . 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 5648	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((𝑔‘𝑥) = 1o ↔ (𝑔‘𝑦) = 1o))
43	42	cbvralvw 2771	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑔‘𝑥) = 1o ↔ ∀𝑦 ∈ (𝑋 ∩ 𝐴)(𝑔‘𝑦) = 1o)
44		1n0 6599	. . . . . . . . . . . . . . . 16 ⊢ 1o ≠ ∅
45	44	neii 2404	. . . . . . . . . . . . . . 15 ⊢ ¬ 1o = ∅
46		eqeq1 2238	. . . . . . . . . . . . . . 15 ⊢ ((𝑔‘𝑦) = 1o → ((𝑔‘𝑦) = ∅ ↔ 1o = ∅))
47	45, 46	mtbiri 681	. . . . . . . . . . . . . 14 ⊢ ((𝑔‘𝑦) = 1o → ¬ (𝑔‘𝑦) = ∅)
48	47	neqned 2409	. . . . . . . . . . . . 13 ⊢ ((𝑔‘𝑦) = 1o → (𝑔‘𝑦) ≠ ∅)
49	48	ralimi 2595	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ (𝑋 ∩ 𝐴)(𝑔‘𝑦) = 1o → ∀𝑦 ∈ (𝑋 ∩ 𝐴)(𝑔‘𝑦) ≠ ∅)
50	43, 49	sylbi 121	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑔‘𝑥) = 1o → ∀𝑦 ∈ (𝑋 ∩ 𝐴)(𝑔‘𝑦) ≠ ∅)
51		fveqeq2 5648	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((𝑔‘𝑥) = ∅ ↔ (𝑔‘𝑦) = ∅))
52	51	cbvralvw 2771	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑔‘𝑥) = ∅ ↔ ∀𝑦 ∈ (𝑋 ∖ 𝐴)(𝑔‘𝑦) = ∅)
53	52	biimpi 120	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑔‘𝑥) = ∅ → ∀𝑦 ∈ (𝑋 ∖ 𝐴)(𝑔‘𝑦) = ∅)
54	50, 53	anim12i 338	. . . . . . . . . 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 2629	. . . . . . . . 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 121	. . . . . . . 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 3292	. . . . . . . 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 277	. . . . . 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 16405	. . . . 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 115	. . . 4 ⊢ (𝜑 → (∃𝑓 ∈ (2o ↑𝑚 𝑋)(∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝑓‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝑓‘𝑥) = ∅) → ∀𝑦 ∈ 𝑋 DECID ¬ 𝑦 ∈ 𝐴))
62		eleq1w 2292	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
63	62	notbid 673	. . . . . 6 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝐴 ↔ ¬ 𝑦 ∈ 𝐴))
64	63	dcbid 845	. . . . 5 ⊢ (𝑥 = 𝑦 → (DECID ¬ 𝑥 ∈ 𝐴 ↔ DECID ¬ 𝑦 ∈ 𝐴))
65	64	cbvralvw 2771	. . . 4 ⊢ (∀𝑥 ∈ 𝑋 DECID ¬ 𝑥 ∈ 𝐴 ↔ ∀𝑦 ∈ 𝑋 DECID ¬ 𝑦 ∈ 𝐴)
66	61, 65	imbitrrdi 162	. . 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 2620	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → STAB 𝑥 ∈ 𝐴)
69		stdcn 854	. . . . 5 ⊢ (STAB 𝑥 ∈ 𝐴 ↔ (DECID ¬ 𝑥 ∈ 𝐴 → DECID 𝑥 ∈ 𝐴))
70	68, 69	sylib 122	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (DECID ¬ 𝑥 ∈ 𝐴 → DECID 𝑥 ∈ 𝐴))
71	70	ralimdva 2599	. . 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 129	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 104   ↔ wb 105  STAB wstab 837  DECID wdc 841   = wceq 1397   ∈ wcel 2202   ≠ wne 2402  ∀wral 2510  ∃wrex 2511  Vcvv 2802   ∖ cdif 3197   ∩ cin 3199   ⊆ wss 3200  ∅c0 3494  ifcif 3605   ↦ cmpt 4150  Oncon0 4460  ωcom 4688  ⟶wf 5322  ‘cfv 5326  (class class class)co 6017  1_oc1o 6574  2_oc2o 6575   ↑_𝑚 cmap 6816

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1o 6581  df-2o 6582  df-map 6818

This theorem is referenced by: (None)