Theorem ssfirab 7196

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 2804	. . 3 ⊢ (𝑤 = ∅ → {𝑥 ∈ 𝑤 ∣ 𝜓} = {𝑥 ∈ ∅ ∣ 𝜓})
2	1	eleq1d 2301	. 2 ⊢ (𝑤 = ∅ → ({𝑥 ∈ 𝑤 ∣ 𝜓} ∈ Fin ↔ {𝑥 ∈ ∅ ∣ 𝜓} ∈ Fin))
3		rabeq 2804	. . 3 ⊢ (𝑤 = 𝑦 → {𝑥 ∈ 𝑤 ∣ 𝜓} = {𝑥 ∈ 𝑦 ∣ 𝜓})
4	3	eleq1d 2301	. 2 ⊢ (𝑤 = 𝑦 → ({𝑥 ∈ 𝑤 ∣ 𝜓} ∈ Fin ↔ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin))
5		rabeq 2804	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → {𝑥 ∈ 𝑤 ∣ 𝜓} = {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓})
6	5	eleq1d 2301	. 2 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ({𝑥 ∈ 𝑤 ∣ 𝜓} ∈ Fin ↔ {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓} ∈ Fin))
7		rabeq 2804	. . 3 ⊢ (𝑤 = 𝐴 → {𝑥 ∈ 𝑤 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜓})
8	7	eleq1d 2301	. 2 ⊢ (𝑤 = 𝐴 → ({𝑥 ∈ 𝑤 ∣ 𝜓} ∈ Fin ↔ {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ Fin))
9		rab0 3536	. . . 4 ⊢ {𝑥 ∈ ∅ ∣ 𝜓} = ∅
10		0fi 7140	. . . 4 ⊢ ∅ ∈ Fin
11	9, 10	eqeltri 2305	. . 3 ⊢ {𝑥 ∈ ∅ ∣ 𝜓} ∈ Fin
12	11	a1i 9	. 2 ⊢ (𝜑 → {𝑥 ∈ ∅ ∣ 𝜓} ∈ Fin)
13		rabun2 3499	. . . . 5 ⊢ {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓} = ({𝑥 ∈ 𝑦 ∣ 𝜓} ∪ {𝑥 ∈ {𝑧} ∣ 𝜓})
14		sbsbc 3045	. . . . . . . . . 10 ⊢ ([𝑧 / 𝑥]𝜓 ↔ [𝑧 / 𝑥]𝜓)
15		vex 2815	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
16		ralsns 3726	. . . . . . . . . . 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 2720	. . . . . . . . 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 3372	. . . . . 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 529	. . . . . . 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 533	. . . . . . . . . 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 3222	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → ¬ 𝑧 ∈ 𝑦)
28		elrabi 2969	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∈ 𝑦 ∣ 𝜓} → 𝑧 ∈ 𝑦)
29	27, 28	nsyl 633	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → ¬ 𝑧 ∈ {𝑥 ∈ 𝑦 ∣ 𝜓})
30		unsnfi 7178	. . . . . . 7 ⊢ (({𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin ∧ 𝑧 ∈ V ∧ ¬ 𝑧 ∈ {𝑥 ∈ 𝑦 ∣ 𝜓}) → ({𝑥 ∈ 𝑦 ∣ 𝜓} ∪ {𝑧}) ∈ Fin)
31	23, 24, 29, 30	syl3anc 1274	. . . . . 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 2310	. . . . 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 2319	. . . 4 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓} ∈ Fin)
34		ralsns 3726	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ V → (∀𝑥 ∈ {𝑧} ¬ 𝜓 ↔ [𝑧 / 𝑥] ¬ 𝜓))
35	15, 34	ax-mp 5	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ {𝑧} ¬ 𝜓 ↔ [𝑧 / 𝑥] ¬ 𝜓)
36		sbsbc 3045	. . . . . . . . . . 11 ⊢ ([𝑧 / 𝑥] ¬ 𝜓 ↔ [𝑧 / 𝑥] ¬ 𝜓)
37		sbn 2006	. . . . . . . . . . 11 ⊢ ([𝑧 / 𝑥] ¬ 𝜓 ↔ ¬ [𝑧 / 𝑥]𝜓)
38	35, 36, 37	3bitr2ri 209	. . . . . . . . . 10 ⊢ (¬ [𝑧 / 𝑥]𝜓 ↔ ∀𝑥 ∈ {𝑧} ¬ 𝜓)
39		rabeq0 3537	. . . . . . . . . 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 3372	. . . . . . 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 3541	. . . . . . 7 ⊢ ({𝑥 ∈ 𝑦 ∣ 𝜓} ∪ ∅) = {𝑥 ∈ 𝑦 ∣ 𝜓}
44	42, 43	eqtrdi 2281	. . . . . 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 2277	. . . . 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 529	. . . . 5 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin) ∧ ¬ [𝑧 / 𝑥]𝜓) → {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin)
47	45, 46	eqeltrd 2309	. . . 4 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin) ∧ ¬ [𝑧 / 𝑥]𝜓) → {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓} ∈ Fin)
48		simplrr 538	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin) → 𝑧 ∈ (𝐴 ∖ 𝑦))
49	48	eldifad 3221	. . . . . 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 1993	. . . . . . . 8 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜓
53	52	nfdc 1707	. . . . . . 7 ⊢ Ⅎ𝑥DECID [𝑧 / 𝑥]𝜓
54		sbequ12 1820	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝜓 ↔ [𝑧 / 𝑥]𝜓))
55	54	dcbid 846	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (DECID 𝜓 ↔ DECID [𝑧 / 𝑥]𝜓))
56	53, 55	rspc 2914	. . . . . 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 844	. . . . 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 806	. . 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 7148	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716  DECID wdc 842   = wceq 1398  [wsb 1811   ∈ wcel 2203  ∀wral 2520  {crab 2524  Vcvv 2812  [wsbc 3041   ∖ cdif 3207   ∪ cun 3208   ⊆ wss 3210  ∅c0 3507  {csn 3688  Fincfn 6974

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-1o 6646  df-er 6766  df-en 6975  df-fin 6977

This theorem is referenced by:  ssfidc  7197  hashfibclem  11199  phivalfi  12902  hashdvds  12911  phiprmpw  12912  phimullem  12915  hashgcdeq  12930  ballotfilemofi  13131  lgsquadlemofi  15936  lgsquadlem1  15937  lgsquadlem2  15938  vtxedgfi  16271  vtxlpfi  16272  konigsberglem5  16474