Theorem ssfirab 6933

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 2730	. . 3 ⊢ (𝑤 = ∅ → {𝑥 ∈ 𝑤 ∣ 𝜓} = {𝑥 ∈ ∅ ∣ 𝜓})
2	1	eleq1d 2246	. 2 ⊢ (𝑤 = ∅ → ({𝑥 ∈ 𝑤 ∣ 𝜓} ∈ Fin ↔ {𝑥 ∈ ∅ ∣ 𝜓} ∈ Fin))
3		rabeq 2730	. . 3 ⊢ (𝑤 = 𝑦 → {𝑥 ∈ 𝑤 ∣ 𝜓} = {𝑥 ∈ 𝑦 ∣ 𝜓})
4	3	eleq1d 2246	. 2 ⊢ (𝑤 = 𝑦 → ({𝑥 ∈ 𝑤 ∣ 𝜓} ∈ Fin ↔ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin))
5		rabeq 2730	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → {𝑥 ∈ 𝑤 ∣ 𝜓} = {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓})
6	5	eleq1d 2246	. 2 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ({𝑥 ∈ 𝑤 ∣ 𝜓} ∈ Fin ↔ {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓} ∈ Fin))
7		rabeq 2730	. . 3 ⊢ (𝑤 = 𝐴 → {𝑥 ∈ 𝑤 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜓})
8	7	eleq1d 2246	. 2 ⊢ (𝑤 = 𝐴 → ({𝑥 ∈ 𝑤 ∣ 𝜓} ∈ Fin ↔ {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ Fin))
9		rab0 3452	. . . 4 ⊢ {𝑥 ∈ ∅ ∣ 𝜓} = ∅
10		0fin 6884	. . . 4 ⊢ ∅ ∈ Fin
11	9, 10	eqeltri 2250	. . 3 ⊢ {𝑥 ∈ ∅ ∣ 𝜓} ∈ Fin
12	11	a1i 9	. 2 ⊢ (𝜑 → {𝑥 ∈ ∅ ∣ 𝜓} ∈ Fin)
13		rabun2 3415	. . . . 5 ⊢ {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓} = ({𝑥 ∈ 𝑦 ∣ 𝜓} ∪ {𝑥 ∈ {𝑧} ∣ 𝜓})
14		sbsbc 2967	. . . . . . . . . 10 ⊢ ([𝑧 / 𝑥]𝜓 ↔ [𝑧 / 𝑥]𝜓)
15		vex 2741	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
16		ralsns 3631	. . . . . . . . . . 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 2654	. . . . . . . . 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 3290	. . . . . 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 3142	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → ¬ 𝑧 ∈ 𝑦)
28		elrabi 2891	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥 ∈ 𝑦 ∣ 𝜓} → 𝑧 ∈ 𝑦)
29	27, 28	nsyl 628	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → ¬ 𝑧 ∈ {𝑥 ∈ 𝑦 ∣ 𝜓})
30		unsnfi 6918	. . . . . . 7 ⊢ (({𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin ∧ 𝑧 ∈ V ∧ ¬ 𝑧 ∈ {𝑥 ∈ 𝑦 ∣ 𝜓}) → ({𝑥 ∈ 𝑦 ∣ 𝜓} ∪ {𝑧}) ∈ Fin)
31	23, 24, 29, 30	syl3anc 1238	. . . . . 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 2255	. . . . 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 2264	. . . 4 ⊢ (((((𝜑 ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) ∧ {𝑥 ∈ 𝑦 ∣ 𝜓} ∈ Fin) ∧ [𝑧 / 𝑥]𝜓) → {𝑥 ∈ (𝑦 ∪ {𝑧}) ∣ 𝜓} ∈ Fin)
34		ralsns 3631	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ V → (∀𝑥 ∈ {𝑧} ¬ 𝜓 ↔ [𝑧 / 𝑥] ¬ 𝜓))
35	15, 34	ax-mp 5	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ {𝑧} ¬ 𝜓 ↔ [𝑧 / 𝑥] ¬ 𝜓)
36		sbsbc 2967	. . . . . . . . . . 11 ⊢ ([𝑧 / 𝑥] ¬ 𝜓 ↔ [𝑧 / 𝑥] ¬ 𝜓)
37		sbn 1952	. . . . . . . . . . 11 ⊢ ([𝑧 / 𝑥] ¬ 𝜓 ↔ ¬ [𝑧 / 𝑥]𝜓)
38	35, 36, 37	3bitr2ri 209	. . . . . . . . . 10 ⊢ (¬ [𝑧 / 𝑥]𝜓 ↔ ∀𝑥 ∈ {𝑧} ¬ 𝜓)
39		rabeq0 3453	. . . . . . . . . 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 3290	. . . . . . 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 3457	. . . . . . 7 ⊢ ({𝑥 ∈ 𝑦 ∣ 𝜓} ∪ ∅) = {𝑥 ∈ 𝑦 ∣ 𝜓}
44	42, 43	eqtrdi 2226	. . . . . 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 2222	. . . . 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 2254	. . . 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 3141	. . . . . 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 1939	. . . . . . . 8 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜓
53	52	nfdc 1659	. . . . . . 7 ⊢ Ⅎ𝑥DECID [𝑧 / 𝑥]𝜓
54		sbequ12 1771	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝜓 ↔ [𝑧 / 𝑥]𝜓))
55	54	dcbid 838	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (DECID 𝜓 ↔ DECID [𝑧 / 𝑥]𝜓))
56	53, 55	rspc 2836	. . . . . 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 836	. . . . 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 798	. . 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 6892	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 708  DECID wdc 834   = wceq 1353  [wsb 1762   ∈ wcel 2148  ∀wral 2455  {crab 2459  Vcvv 2738  [wsbc 2963   ∖ cdif 3127   ∪ cun 3128   ⊆ wss 3130  ∅c0 3423  {csn 3593  Fincfn 6740

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-1o 6417  df-er 6535  df-en 6741  df-fin 6743

This theorem is referenced by:  ssfidc  6934  phivalfi  12212  hashdvds  12221  phiprmpw  12222  phimullem  12225  hashgcdeq  12239