Theorem findcard2 9089

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 5310. (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 8912	. . 3 ⊢ (𝑥 ∈ Fin ↔ ∃𝑤 ∈ ω 𝑥 ≈ 𝑤)
3		breq2 5102	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ ∅))
4	3	imbi1d 341	. . . . . . 7 ⊢ (𝑤 = ∅ → ((𝑥 ≈ 𝑤 → 𝜑) ↔ (𝑥 ≈ ∅ → 𝜑)))
5	4	albidv 1921	. . . . . 6 ⊢ (𝑤 = ∅ → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ ∅ → 𝜑)))
6		breq2 5102	. . . . . . . 8 ⊢ (𝑤 = 𝑣 → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ 𝑣))
7	6	imbi1d 341	. . . . . . 7 ⊢ (𝑤 = 𝑣 → ((𝑥 ≈ 𝑤 → 𝜑) ↔ (𝑥 ≈ 𝑣 → 𝜑)))
8	7	albidv 1921	. . . . . 6 ⊢ (𝑤 = 𝑣 → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ 𝑣 → 𝜑)))
9		breq2 5102	. . . . . . . 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 1921	. . . . . 6 ⊢ (𝑤 = suc 𝑣 → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ suc 𝑣 → 𝜑)))
12		en0 8955	. . . . . . . 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 1796	. . . . . 6 ⊢ ∀𝑥(𝑥 ≈ ∅ → 𝜑)
18		nnon 7814	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ ω → 𝑣 ∈ On)
19		rexdif1en 9085	. . . . . . . . . . . 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 4764	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑤 → {𝑧} ⊆ 𝑤)
22		uncom 4110	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∖ {𝑧}) ∪ {𝑧}) = ({𝑧} ∪ (𝑤 ∖ {𝑧}))
23		undif 4434	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑧} ⊆ 𝑤 ↔ ({𝑧} ∪ (𝑤 ∖ {𝑧})) = 𝑤)
24	23	biimpi 216	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑧} ⊆ 𝑤 → ({𝑧} ∪ (𝑤 ∖ {𝑧})) = 𝑤)
25	22, 24	eqtrid 2783	. . . . . . . . . . . . . . . 16 ⊢ ({𝑧} ⊆ 𝑤 → ((𝑤 ∖ {𝑧}) ∪ {𝑧}) = 𝑤)
26		vex 3444	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑤 ∈ V
27	26	difexi 5275	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∖ {𝑧}) ∈ V
28		breq1 5101	. . . . . . . . . . . . . . . . . . . 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 4113	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = (𝑤 ∖ {𝑧}) → (𝑦 ∪ {𝑧}) = ((𝑤 ∖ {𝑧}) ∪ {𝑧}))
31	30	sbceq1d 3745	. . . . . . . . . . . . . . . . . . . 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 5101	. . . . . . . . . . . . . . . . . . . . . 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 1989	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → (𝑦 ≈ 𝑣 → 𝜒))
38		rspe 3226	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → ∃𝑣 ∈ ω 𝑦 ≈ 𝑣)
39		isfi 8912	. . . . . . . . . . . . . . . . . . . . . 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 3444	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑦 ∈ V
47		vsnex 5379	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {𝑧} ∈ V
48	46, 47	unex 7689	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∪ {𝑧}) ∈ V
49		findcard2.3	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝜑 ↔ 𝜃))
50	48, 49	sbcie 3782	. . . . . . . . . . . . . . . . . . 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 3515	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ∈ ω ∧ (𝑤 ∖ {𝑧}) ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [((𝑤 ∖ {𝑧}) ∪ {𝑧}) / 𝑥]𝜑))
53		dfsbcq 3742	. . . . . . . . . . . . . . . . . 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 3135	. . . . . . . . . . . 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 1930	. . . . . . 7 ⊢ (𝑣 ∈ ω → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → ∀𝑤(𝑤 ≈ suc 𝑣 → [𝑤 / 𝑥]𝜑)))
65		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑤(𝑥 ≈ suc 𝑣 → 𝜑)
66		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑤 ≈ suc 𝑣
67		nfsbc1v 3760	. . . . . . . . 9 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝜑
68	66, 67	nfim 1897	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑤 ≈ suc 𝑣 → [𝑤 / 𝑥]𝜑)
69		breq1 5101	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑥 ≈ suc 𝑣 ↔ 𝑤 ≈ suc 𝑣))
70		sbceq1a 3751	. . . . . . . . 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 2345	. . . . . . 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 7841	. . . . 5 ⊢ (𝑤 ∈ ω → ∀𝑥(𝑥 ≈ 𝑤 → 𝜑))
75	74	19.21bi 2196	. . . 4 ⊢ (𝑤 ∈ ω → (𝑥 ≈ 𝑤 → 𝜑))
76	75	rexlimiv 3130	. . 3 ⊢ (∃𝑤 ∈ ω 𝑥 ≈ 𝑤 → 𝜑)
77	2, 76	sylbi 217	. 2 ⊢ (𝑥 ∈ Fin → 𝜑)
78	1, 77	vtoclga 3532	1 ⊢ (𝐴 ∈ Fin → 𝜏)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539   = wceq 1541   ∈ wcel 2113  ∃wrex 3060  [wsbc 3740   ∖ cdif 3898   ∪ cun 3899   ⊆ wss 3901  ∅c0 4285  {csn 4580   class class class wbr 5098  Oncon0 6317  suc csuc 6319  ωcom 7808   ≈ cen 8880  Fincfn 8883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-om 7809  df-en 8884  df-fin 8887

This theorem is referenced by:  findcard2s  9090  ssfi  9097  cnvfi  9100  fnfi  9102  frfi  9185  imafiOLD  9216  pwfi  9219  iunfi  9243  finsschain  9259  infdiffi  9567  fin1a2lem10  10319  wunfi  10632  rexfiuz  15271  modfsummod  15717  lcmfunsnlem  16568  lcmfun  16572  drsdirfi  18228  fiuncmp  23348  finiunmbl  25501  fineqvac  35272  mbfresfi  37867  heibor1lem  38010  pclfinclN  40210