Theorem findcard2 9204

Description: Schema for induction on the cardinality of a finite set. The inductive step shows that the result is true if one more element is added to the set. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 8-Jul-2010.) Avoid ax-pow 5365. (Revised by BTernaryTau, 26-Aug-2024.)

Hypotheses

Ref	Expression
findcard2.1	⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
findcard2.2	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
findcard2.3	⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝜑 ↔ 𝜃))
findcard2.4	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
findcard2.5	⊢ 𝜓
findcard2.6	⊢ (𝑦 ∈ Fin → (𝜒 → 𝜃))

Assertion

Ref	Expression
findcard2	⊢ (𝐴 ∈ Fin → 𝜏)

Distinct variable groups:   𝜒,𝑥   𝜃,𝑥   𝜏,𝑥   𝑥,𝐴   𝜑,𝑦,𝑧   𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦,𝑧)   𝜒(𝑦,𝑧)   𝜃(𝑦,𝑧)   𝜏(𝑦,𝑧)   𝐴(𝑦,𝑧)

Proof of Theorem findcard2

Dummy variables 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		findcard2.4	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
2		isfi 9016	. . 3 ⊢ (𝑥 ∈ Fin ↔ ∃𝑤 ∈ ω 𝑥 ≈ 𝑤)
3		breq2 5147	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ ∅))
4	3	imbi1d 341	. . . . . . 7 ⊢ (𝑤 = ∅ → ((𝑥 ≈ 𝑤 → 𝜑) ↔ (𝑥 ≈ ∅ → 𝜑)))
5	4	albidv 1920	. . . . . 6 ⊢ (𝑤 = ∅ → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ ∅ → 𝜑)))
6		breq2 5147	. . . . . . . 8 ⊢ (𝑤 = 𝑣 → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ 𝑣))
7	6	imbi1d 341	. . . . . . 7 ⊢ (𝑤 = 𝑣 → ((𝑥 ≈ 𝑤 → 𝜑) ↔ (𝑥 ≈ 𝑣 → 𝜑)))
8	7	albidv 1920	. . . . . 6 ⊢ (𝑤 = 𝑣 → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ 𝑣 → 𝜑)))
9		breq2 5147	. . . . . . . 8 ⊢ (𝑤 = suc 𝑣 → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ suc 𝑣))
10	9	imbi1d 341	. . . . . . 7 ⊢ (𝑤 = suc 𝑣 → ((𝑥 ≈ 𝑤 → 𝜑) ↔ (𝑥 ≈ suc 𝑣 → 𝜑)))
11	10	albidv 1920	. . . . . 6 ⊢ (𝑤 = suc 𝑣 → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ suc 𝑣 → 𝜑)))
12		en0 9058	. . . . . . . 8 ⊢ (𝑥 ≈ ∅ ↔ 𝑥 = ∅)
13		findcard2.5	. . . . . . . . 9 ⊢ 𝜓
14		findcard2.1	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
15	13, 14	mpbiri 258	. . . . . . . 8 ⊢ (𝑥 = ∅ → 𝜑)
16	12, 15	sylbi 217	. . . . . . 7 ⊢ (𝑥 ≈ ∅ → 𝜑)
17	16	ax-gen 1795	. . . . . 6 ⊢ ∀𝑥(𝑥 ≈ ∅ → 𝜑)
18		nnon 7893	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ ω → 𝑣 ∈ On)
19		rexdif1en 9198	. . . . . . . . . . . 12 ⊢ ((𝑣 ∈ On ∧ 𝑤 ≈ suc 𝑣) → ∃𝑧 ∈ 𝑤 (𝑤 ∖ {𝑧}) ≈ 𝑣)
20	18, 19	sylan 580	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣) → ∃𝑧 ∈ 𝑤 (𝑤 ∖ {𝑧}) ≈ 𝑣)
21		snssi 4808	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑤 → {𝑧} ⊆ 𝑤)
22		uncom 4158	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∖ {𝑧}) ∪ {𝑧}) = ({𝑧} ∪ (𝑤 ∖ {𝑧}))
23		undif 4482	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑧} ⊆ 𝑤 ↔ ({𝑧} ∪ (𝑤 ∖ {𝑧})) = 𝑤)
24	23	biimpi 216	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑧} ⊆ 𝑤 → ({𝑧} ∪ (𝑤 ∖ {𝑧})) = 𝑤)
25	22, 24	eqtrid 2789	. . . . . . . . . . . . . . . 16 ⊢ ({𝑧} ⊆ 𝑤 → ((𝑤 ∖ {𝑧}) ∪ {𝑧}) = 𝑤)
26		vex 3484	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑤 ∈ V
27	26	difexi 5330	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∖ {𝑧}) ∈ V
28		breq1 5146	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑤 ∖ {𝑧}) → (𝑦 ≈ 𝑣 ↔ (𝑤 ∖ {𝑧}) ≈ 𝑣))
29	28	anbi2d 630	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝑤 ∖ {𝑧}) → ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) ↔ (𝑣 ∈ ω ∧ (𝑤 ∖ {𝑧}) ≈ 𝑣)))
30		uneq1 4161	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = (𝑤 ∖ {𝑧}) → (𝑦 ∪ {𝑧}) = ((𝑤 ∖ {𝑧}) ∪ {𝑧}))
31	30	sbceq1d 3793	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑤 ∖ {𝑧}) → ([(𝑦 ∪ {𝑧}) / 𝑥]𝜑 ↔ [((𝑤 ∖ {𝑧}) ∪ {𝑧}) / 𝑥]𝜑))
32	31	imbi2d 340	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝑤 ∖ {𝑧}) → ((∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [(𝑦 ∪ {𝑧}) / 𝑥]𝜑) ↔ (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [((𝑤 ∖ {𝑧}) ∪ {𝑧}) / 𝑥]𝜑)))
33	29, 32	imbi12d 344	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝑤 ∖ {𝑧}) → (((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [(𝑦 ∪ {𝑧}) / 𝑥]𝜑)) ↔ ((𝑣 ∈ ω ∧ (𝑤 ∖ {𝑧}) ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [((𝑤 ∖ {𝑧}) ∪ {𝑧}) / 𝑥]𝜑))))
34		breq1 5146	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝑣 ↔ 𝑦 ≈ 𝑣))
35		findcard2.2	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
36	34, 35	imbi12d 344	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑦 → ((𝑥 ≈ 𝑣 → 𝜑) ↔ (𝑦 ≈ 𝑣 → 𝜒)))
37	36	spvv 1996	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → (𝑦 ≈ 𝑣 → 𝜒))
38		rspe 3249	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → ∃𝑣 ∈ ω 𝑦 ≈ 𝑣)
39		isfi 9016	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ Fin ↔ ∃𝑣 ∈ ω 𝑦 ≈ 𝑣)
40	38, 39	sylibr 234	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → 𝑦 ∈ Fin)
41		pm2.27 42	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ≈ 𝑣 → ((𝑦 ≈ 𝑣 → 𝜒) → 𝜒))
42	41	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → ((𝑦 ≈ 𝑣 → 𝜒) → 𝜒))
43		findcard2.6	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ Fin → (𝜒 → 𝜃))
44	40, 42, 43	sylsyld 61	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → ((𝑦 ≈ 𝑣 → 𝜒) → 𝜃))
45	37, 44	syl5 34	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → 𝜃))
46		vex 3484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑦 ∈ V
47		vsnex 5434	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {𝑧} ∈ V
48	46, 47	unex 7764	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∪ {𝑧}) ∈ V
49		findcard2.3	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝜑 ↔ 𝜃))
50	48, 49	sbcie 3830	. . . . . . . . . . . . . . . . . . 19 ⊢ ([(𝑦 ∪ {𝑧}) / 𝑥]𝜑 ↔ 𝜃)
51	45, 50	imbitrrdi 252	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [(𝑦 ∪ {𝑧}) / 𝑥]𝜑))
52	27, 33, 51	vtocl 3558	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ∈ ω ∧ (𝑤 ∖ {𝑧}) ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [((𝑤 ∖ {𝑧}) ∪ {𝑧}) / 𝑥]𝜑))
53		dfsbcq 3790	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∖ {𝑧}) ∪ {𝑧}) = 𝑤 → ([((𝑤 ∖ {𝑧}) ∪ {𝑧}) / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝜑))
54	53	imbi2d 340	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∖ {𝑧}) ∪ {𝑧}) = 𝑤 → ((∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [((𝑤 ∖ {𝑧}) ∪ {𝑧}) / 𝑥]𝜑) ↔ (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑)))
55	52, 54	imbitrid 244	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 ∖ {𝑧}) ∪ {𝑧}) = 𝑤 → ((𝑣 ∈ ω ∧ (𝑤 ∖ {𝑧}) ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑)))
56	21, 25, 55	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑤 → ((𝑣 ∈ ω ∧ (𝑤 ∖ {𝑧}) ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑)))
57	56	expd 415	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑤 → (𝑣 ∈ ω → ((𝑤 ∖ {𝑧}) ≈ 𝑣 → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑))))
58	57	com12 32	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ ω → (𝑧 ∈ 𝑤 → ((𝑤 ∖ {𝑧}) ≈ 𝑣 → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑))))
59	58	rexlimdv 3153	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ ω → (∃𝑧 ∈ 𝑤 (𝑤 ∖ {𝑧}) ≈ 𝑣 → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑)))
60	59	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣) → (∃𝑧 ∈ 𝑤 (𝑤 ∖ {𝑧}) ≈ 𝑣 → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑)))
61	20, 60	mpd 15	. . . . . . . . . 10 ⊢ ((𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑))
62	61	ex 412	. . . . . . . . 9 ⊢ (𝑣 ∈ ω → (𝑤 ≈ suc 𝑣 → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑)))
63	62	com23 86	. . . . . . . 8 ⊢ (𝑣 ∈ ω → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → (𝑤 ≈ suc 𝑣 → [𝑤 / 𝑥]𝜑)))
64	63	alrimdv 1929	. . . . . . 7 ⊢ (𝑣 ∈ ω → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → ∀𝑤(𝑤 ≈ suc 𝑣 → [𝑤 / 𝑥]𝜑)))
65		nfv 1914	. . . . . . . 8 ⊢ Ⅎ𝑤(𝑥 ≈ suc 𝑣 → 𝜑)
66		nfv 1914	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑤 ≈ suc 𝑣
67		nfsbc1v 3808	. . . . . . . . 9 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝜑
68	66, 67	nfim 1896	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑤 ≈ suc 𝑣 → [𝑤 / 𝑥]𝜑)
69		breq1 5146	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑥 ≈ suc 𝑣 ↔ 𝑤 ≈ suc 𝑣))
70		sbceq1a 3799	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑥]𝜑))
71	69, 70	imbi12d 344	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → ((𝑥 ≈ suc 𝑣 → 𝜑) ↔ (𝑤 ≈ suc 𝑣 → [𝑤 / 𝑥]𝜑)))
72	65, 68, 71	cbvalv1 2343	. . . . . . 7 ⊢ (∀𝑥(𝑥 ≈ suc 𝑣 → 𝜑) ↔ ∀𝑤(𝑤 ≈ suc 𝑣 → [𝑤 / 𝑥]𝜑))
73	64, 72	imbitrrdi 252	. . . . . 6 ⊢ (𝑣 ∈ ω → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → ∀𝑥(𝑥 ≈ suc 𝑣 → 𝜑)))
74	5, 8, 11, 17, 73	finds1 7921	. . . . 5 ⊢ (𝑤 ∈ ω → ∀𝑥(𝑥 ≈ 𝑤 → 𝜑))
75	74	19.21bi 2189	. . . 4 ⊢ (𝑤 ∈ ω → (𝑥 ≈ 𝑤 → 𝜑))
76	75	rexlimiv 3148	. . 3 ⊢ (∃𝑤 ∈ ω 𝑥 ≈ 𝑤 → 𝜑)
77	2, 76	sylbi 217	. 2 ⊢ (𝑥 ∈ Fin → 𝜑)
78	1, 77	vtoclga 3577	1 ⊢ (𝐴 ∈ Fin → 𝜏)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2108  ∃wrex 3070  [wsbc 3788   ∖ cdif 3948   ∪ cun 3949   ⊆ wss 3951  ∅c0 4333  {csn 4626   class class class wbr 5143  Oncon0 6384  suc csuc 6386  ωcom 7887   ≈ cen 8982  Fincfn 8985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-en 8986  df-fin 8989

This theorem is referenced by:  findcard2s  9205  ssfi  9213  cnvfi  9216  fnfi  9218  frfi  9321  imafiOLD  9354  pwfi  9357  iunfi  9383  finsschain  9399  infdiffi  9698  fin1a2lem10  10449  wunfi  10761  rexfiuz  15386  modfsummod  15830  lcmfunsnlem  16678  lcmfun  16682  drsdirfi  18351  fiuncmp  23412  finiunmbl  25579  fineqvac  35111  mbfresfi  37673  heibor1lem  37816  pclfinclN  39952