Theorem ssfirab 7006

Description: A subset of a finite set is finite if it is defined by a decidable property. (Contributed by Jim Kingdon, 27-May-2022.)

Hypotheses

Ref	Expression
ssfirab.a	⊢ (𝜑 → 𝐴 ∈ Fin)
ssfirab.dc	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 DECID 𝜓)

Assertion

Ref	Expression
ssfirab	⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ Fin)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem ssfirab

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

Step	Hyp	Ref	Expression
1		rabeq 2755	. . 3 ⊢ (𝑤 = ∅ → {𝑥 ∈ 𝑤 ∣ 𝜓} = {𝑥 ∈ ∅ ∣ 𝜓})
2	1	eleq1d 2265	. 2 ⊢ (𝑤 = ∅ → ({𝑥 ∈ 𝑤 ∣ 𝜓} ∈ Fin ↔ {𝑥 ∈ ∅ ∣ 𝜓} ∈ Fin))
3		rabeq 2755	. . 3 ⊢ (𝑤 = 𝑦 → {𝑥 ∈ 𝑤 ∣ 𝜓} = {𝑥 ∈ 𝑦 ∣ 𝜓})
4	3	eleq1d 2265	. 2 ⊢ (𝑤 = 𝑦 → ({𝑥 ∈ 𝑤 ∣ 𝜓} ∈ Fin ↔ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin))
5		rabeq 2755	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → {𝑥 ∈ 𝑤 ∣ 𝜓} = {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓})
6	5	eleq1d 2265	. 2 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ({𝑥 ∈ 𝑤 ∣ 𝜓} ∈ Fin ↔ {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓} ∈ Fin))
7		rabeq 2755	. . 3 ⊢ (𝑤 = 𝐴 → {𝑥 ∈ 𝑤 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜓})
8	7	eleq1d 2265	. 2 ⊢ (𝑤 = 𝐴 → ({𝑥 ∈ 𝑤 ∣ 𝜓} ∈ Fin ↔ {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ Fin))
9		rab0 3480	. . . 4 ⊢ {𝑥 ∈ ∅ ∣ 𝜓} = ∅
10		0fin 6954	. . . 4 ⊢ ∅ ∈ Fin
11	9, 10	eqeltri 2269	. . 3 ⊢ {𝑥 ∈ ∅ ∣ 𝜓} ∈ Fin
12	11	a1i 9	. 2 ⊢ (𝜑 → {𝑥 ∈ ∅ ∣ 𝜓} ∈ Fin)
13		rabun2 3443	. . . . 5 ⊢ {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓} = ({𝑥 ∈ 𝑦 ∣ 𝜓} ∪ {𝑥 ∈ {𝑧} ∣ 𝜓})
14		sbsbc 2993	. . . . . . . . . 10 ⊢ ([𝑧 / 𝑥]𝜓 ↔ [𝑧 / 𝑥]𝜓)
15		vex 2766	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
16		ralsns 3661	. . . . . . . . . . 11 ⊢ (𝑧 ∈ V → (∀𝑥 ∈ {𝑧}𝜓 ↔ [𝑧 / 𝑥]𝜓))
17	15, 16	ax-mp 5	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ {𝑧}𝜓 ↔ [𝑧 / 𝑥]𝜓)
18	14, 17	bitr4i 187	. . . . . . . . 9 ⊢ ([𝑧 / 𝑥]𝜓 ↔ ∀𝑥 ∈ {𝑧}𝜓)
19		rabid2 2674	. . . . . . . . 9 ⊢ ({𝑧} = {𝑥 ∈ {𝑧} ∣ 𝜓} ↔ ∀𝑥 ∈ {𝑧}𝜓)
20	18, 19	sylbb2 138	. . . . . . . 8 ⊢ ([𝑧 / 𝑥]𝜓 → {𝑧} = {𝑥 ∈ {𝑧} ∣ 𝜓})
21	20	adantl 277	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → {𝑧} = {𝑥 ∈ {𝑧} ∣ 𝜓})
22	21	uneq2d 3318	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → ({𝑥 ∈ 𝑦 ∣ 𝜓} ∪ {𝑧}) = ({𝑥 ∈ 𝑦 ∣ 𝜓} ∪ {𝑥 ∈ {𝑧} ∣ 𝜓}))
23		simplr 528	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin)
24	15	a1i 9	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → 𝑧 ∈ V)
25		simprr 531	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → 𝑧 ∈ (𝐴 ∖ 𝑦))
26	25	ad2antrr 488	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → 𝑧 ∈ (𝐴 ∖ 𝑦))
27	26	eldifbd 3169	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → ¬ 𝑧 ∈ 𝑦)
28		elrabi 2917	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∈ 𝑦 ∣ 𝜓} → 𝑧 ∈ 𝑦)
29	27, 28	nsyl 629	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → ¬ 𝑧 ∈ {𝑥 ∈ 𝑦 ∣ 𝜓})
30		unsnfi 6989	. . . . . . 7 ⊢ (({𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin ∧ 𝑧 ∈ V ∧ ¬ 𝑧 ∈ {𝑥 ∈ 𝑦 ∣ 𝜓}) → ({𝑥 ∈ 𝑦 ∣ 𝜓} ∪ {𝑧}) ∈ Fin)
31	23, 24, 29, 30	syl3anc 1249	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → ({𝑥 ∈ 𝑦 ∣ 𝜓} ∪ {𝑧}) ∈ Fin)
32	22, 31	eqeltrrd 2274	. . . . 5 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → ({𝑥 ∈ 𝑦 ∣ 𝜓} ∪ {𝑥 ∈ {𝑧} ∣ 𝜓}) ∈ Fin)
33	13, 32	eqeltrid 2283	. . . 4 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓} ∈ Fin)
34		ralsns 3661	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ V → (∀𝑥 ∈ {𝑧} ¬ 𝜓 ↔ [𝑧 / 𝑥] ¬ 𝜓))
35	15, 34	ax-mp 5	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ {𝑧} ¬ 𝜓 ↔ [𝑧 / 𝑥] ¬ 𝜓)
36		sbsbc 2993	. . . . . . . . . . 11 ⊢ ([𝑧 / 𝑥] ¬ 𝜓 ↔ [𝑧 / 𝑥] ¬ 𝜓)
37		sbn 1971	. . . . . . . . . . 11 ⊢ ([𝑧 / 𝑥] ¬ 𝜓 ↔ ¬ [𝑧 / 𝑥]𝜓)
38	35, 36, 37	3bitr2ri 209	. . . . . . . . . 10 ⊢ (¬ [𝑧 / 𝑥]𝜓 ↔ ∀𝑥 ∈ {𝑧} ¬ 𝜓)
39		rabeq0 3481	. . . . . . . . . 10 ⊢ ({𝑥 ∈ {𝑧} ∣ 𝜓} = ∅ ↔ ∀𝑥 ∈ {𝑧} ¬ 𝜓)
40	38, 39	sylbb2 138	. . . . . . . . 9 ⊢ (¬ [𝑧 / 𝑥]𝜓 → {𝑥 ∈ {𝑧} ∣ 𝜓} = ∅)
41	40	adantl 277	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin) ∧ ¬ [𝑧 / 𝑥]𝜓) → {𝑥 ∈ {𝑧} ∣ 𝜓} = ∅)
42	41	uneq2d 3318	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin) ∧ ¬ [𝑧 / 𝑥]𝜓) → ({𝑥 ∈ 𝑦 ∣ 𝜓} ∪ {𝑥 ∈ {𝑧} ∣ 𝜓}) = ({𝑥 ∈ 𝑦 ∣ 𝜓} ∪ ∅))
43		un0 3485	. . . . . . 7 ⊢ ({𝑥 ∈ 𝑦 ∣ 𝜓} ∪ ∅) = {𝑥 ∈ 𝑦 ∣ 𝜓}
44	42, 43	eqtrdi 2245	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin) ∧ ¬ [𝑧 / 𝑥]𝜓) → ({𝑥 ∈ 𝑦 ∣ 𝜓} ∪ {𝑥 ∈ {𝑧} ∣ 𝜓}) = {𝑥 ∈ 𝑦 ∣ 𝜓})
45	13, 44	eqtrid 2241	. . . . 5 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin) ∧ ¬ [𝑧 / 𝑥]𝜓) → {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓} = {𝑥 ∈ 𝑦 ∣ 𝜓})
46		simplr 528	. . . . 5 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin) ∧ ¬ [𝑧 / 𝑥]𝜓) → {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin)
47	45, 46	eqeltrd 2273	. . . 4 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin) ∧ ¬ [𝑧 / 𝑥]𝜓) → {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓} ∈ Fin)
48		simplrr 536	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin) → 𝑧 ∈ (𝐴 ∖ 𝑦))
49	48	eldifad 3168	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin) → 𝑧 ∈ 𝐴)
50		ssfirab.dc	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 DECID 𝜓)
51	50	ad3antrrr 492	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin) → ∀𝑥 ∈ 𝐴 DECID 𝜓)
52		nfs1v 1958	. . . . . . . 8 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜓
53	52	nfdc 1673	. . . . . . 7 ⊢ Ⅎ𝑥DECID [𝑧 / 𝑥]𝜓
54		sbequ12 1785	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝜓 ↔ [𝑧 / 𝑥]𝜓))
55	54	dcbid 839	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (DECID 𝜓 ↔ DECID [𝑧 / 𝑥]𝜓))
56	53, 55	rspc 2862	. . . . . 6 ⊢ (𝑧 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 DECID 𝜓 → DECID [𝑧 / 𝑥]𝜓))
57	49, 51, 56	sylc 62	. . . . 5 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin) → DECID [𝑧 / 𝑥]𝜓)
58		exmiddc 837	. . . . 5 ⊢ (DECID [𝑧 / 𝑥]𝜓 → ([𝑧 / 𝑥]𝜓 ∨ ¬ [𝑧 / 𝑥]𝜓))
59	57, 58	syl 14	. . . 4 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin) → ([𝑧 / 𝑥]𝜓 ∨ ¬ [𝑧 / 𝑥]𝜓))
60	33, 47, 59	mpjaodan 799	. . 3 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin) → {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓} ∈ Fin)
61	60	ex 115	. 2 ⊢ (((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → ({𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin → {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓} ∈ Fin))
62		ssfirab.a	. 2 ⊢ (𝜑 → 𝐴 ∈ Fin)
63	2, 4, 6, 8, 12, 61, 62	findcard2sd 6962	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709  DECID wdc 835   = wceq 1364  [wsb 1776   ∈ wcel 2167  ∀wral 2475  {crab 2479  Vcvv 2763  [wsbc 2989   ∖ cdif 3154   ∪ cun 3155   ⊆ wss 3157  ∅c0 3451  {csn 3623  Fincfn 6808

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-1o 6483  df-er 6601  df-en 6809  df-fin 6811

This theorem is referenced by:  ssfidc  7007  phivalfi  12405  hashdvds  12414  phiprmpw  12415  phimullem  12418  hashgcdeq  12433  lgsquadlemofi  15401  lgsquadlem1  15402  lgsquadlem2  15403