Theorem findcard2 9159

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 5330. (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 8981	. . 3 ⊢ (𝑥 ∈ Fin ↔ ∃𝑤 ∈ ω 𝑥 ≈ 𝑤)
3		breq2 5107	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ ∅))
4	3	imbi1d 344	. . . . . . 7 ⊢ (𝑤 = ∅ → ((𝑥 ≈ 𝑤 → 𝜑) ↔ (𝑥 ≈ ∅ → 𝜑)))
5	4	albidv 1953	. . . . . 6 ⊢ (𝑤 = ∅ → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ ∅ → 𝜑)))
6		breq2 5107	. . . . . . . 8 ⊢ (𝑤 = 𝑣 → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ 𝑣))
7	6	imbi1d 344	. . . . . . 7 ⊢ (𝑤 = 𝑣 → ((𝑥 ≈ 𝑤 → 𝜑) ↔ (𝑥 ≈ 𝑣 → 𝜑)))
8	7	albidv 1953	. . . . . 6 ⊢ (𝑤 = 𝑣 → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ 𝑣 → 𝜑)))
9		breq2 5107	. . . . . . . 8 ⊢ (𝑤 = suc 𝑣 → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ suc 𝑣))
10	9	imbi1d 344	. . . . . . 7 ⊢ (𝑤 = suc 𝑣 → ((𝑥 ≈ 𝑤 → 𝜑) ↔ (𝑥 ≈ suc 𝑣 → 𝜑)))
11	10	albidv 1953	. . . . . 6 ⊢ (𝑤 = suc 𝑣 → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ suc 𝑣 → 𝜑)))
12		en0 9024	. . . . . . . 8 ⊢ (𝑥 ≈ ∅ ↔ 𝑥 = ∅)
13		findcard2.5	. . . . . . . . 9 ⊢ 𝜓
14		findcard2.1	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
15	13, 14	mpbiri 261	. . . . . . . 8 ⊢ (𝑥 = ∅ → 𝜑)
16	12, 15	sylbi 220	. . . . . . 7 ⊢ (𝑥 ≈ ∅ → 𝜑)
17	16	ax-gen 1828	. . . . . 6 ⊢ ∀𝑥(𝑥 ≈ ∅ → 𝜑)
18		nnon 7868	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ ω → 𝑣 ∈ On)
19		rexdif1en 9155	. . . . . . . . . . . 12 ⊢ ((𝑣 ∈ On ∧ 𝑤 ≈ suc 𝑣) → ∃𝑧 ∈ 𝑤 (𝑤 ∖ {𝑧}) ≈ 𝑣)
20	18, 19	sylan 592	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣) → ∃𝑧 ∈ 𝑤 (𝑤 ∖ {𝑧}) ≈ 𝑣)
21		snssi 4746	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑤 → {𝑧} ⊆ 𝑤)
22		uncom 4105	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∖ {𝑧}) ∪ {𝑧}) = ({𝑧} ∪ (𝑤 ∖ {𝑧}))
23		undif 4438	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑧} ⊆ 𝑤 ↔ ({𝑧} ∪ (𝑤 ∖ {𝑧})) = 𝑤)
24	23	biimpi 219	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑧} ⊆ 𝑤 → ({𝑧} ∪ (𝑤 ∖ {𝑧})) = 𝑤)
25	22, 24	eqtrid 2807	. . . . . . . . . . . . . . . 16 ⊢ ({𝑧} ⊆ 𝑤 → ((𝑤 ∖ {𝑧}) ∪ {𝑧}) = 𝑤)
26		vex 3454	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑤 ∈ V
27	26	difexi 5295	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∖ {𝑧}) ∈ V
28		breq1 5106	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑤 ∖ {𝑧}) → (𝑦 ≈ 𝑣 ↔ (𝑤 ∖ {𝑧}) ≈ 𝑣))
29	28	anbi2d 642	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝑤 ∖ {𝑧}) → ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) ↔ (𝑣 ∈ ω ∧ (𝑤 ∖ {𝑧}) ≈ 𝑣)))
30		uneq1 4108	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = (𝑤 ∖ {𝑧}) → (𝑦 ∪ {𝑧}) = ((𝑤 ∖ {𝑧}) ∪ {𝑧}))
31	30	sbceq1d 3744	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑤 ∖ {𝑧}) → ([(𝑦 ∪ {𝑧}) / 𝑥]𝜑 ↔ [((𝑤 ∖ {𝑧}) ∪ {𝑧}) / 𝑥]𝜑))
32	31	imbi2d 343	. . . . . . . . . . . . . . . . . . 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 347	. . . . . . . . . . . . . . . . . 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 5106	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝑣 ↔ 𝑦 ≈ 𝑣))
35		findcard2.2	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
36	34, 35	imbi12d 347	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑦 → ((𝑥 ≈ 𝑣 → 𝜑) ↔ (𝑦 ≈ 𝑣 → 𝜒)))
37	36	spvv 2021	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → (𝑦 ≈ 𝑣 → 𝜒))
38		rspe 3252	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → ∃𝑣 ∈ ω 𝑦 ≈ 𝑣)
39		isfi 8981	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ Fin ↔ ∃𝑣 ∈ ω 𝑦 ≈ 𝑣)
40	38, 39	sylibr 237	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → 𝑦 ∈ Fin)
41		pm2.27 43	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ≈ 𝑣 → ((𝑦 ≈ 𝑣 → 𝜒) → 𝜒))
42	41	adantl 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → ((𝑦 ≈ 𝑣 → 𝜒) → 𝜒))
43		findcard2.6	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ Fin → (𝜒 → 𝜃))
44	40, 42, 43	sylsyld 62	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → ((𝑦 ≈ 𝑣 → 𝜒) → 𝜃))
45	37, 44	syl5 35	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → 𝜃))
46		vex 3454	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑦 ∈ V
47		vsnex 5400	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {𝑧} ∈ V
48	46, 47	unex 7746	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∪ {𝑧}) ∈ V
49		findcard2.3	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝜑 ↔ 𝜃))
50	48, 49	sbcie 3780	. . . . . . . . . . . . . . . . . . 19 ⊢ ([(𝑦 ∪ {𝑧}) / 𝑥]𝜑 ↔ 𝜃)
51	45, 50	imbitrrdi 255	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [(𝑦 ∪ {𝑧}) / 𝑥]𝜑))
52	27, 33, 51	vtocl 3520	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ∈ ω ∧ (𝑤 ∖ {𝑧}) ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [((𝑤 ∖ {𝑧}) ∪ {𝑧}) / 𝑥]𝜑))
53		dfsbcq 3741	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ∖ {𝑧}) ∪ {𝑧}) = 𝑤 → ([((𝑤 ∖ {𝑧}) ∪ {𝑧}) / 𝑥]𝜑 ↔ [𝑤 / 𝑥]𝜑))
54	53	imbi2d 343	. . . . . . . . . . . . . . . . 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 247	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 ∖ {𝑧}) ∪ {𝑧}) = 𝑤 → ((𝑣 ∈ ω ∧ (𝑤 ∖ {𝑧}) ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑)))
56	21, 25, 55	3syl 19	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑤 → ((𝑣 ∈ ω ∧ (𝑤 ∖ {𝑧}) ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑)))
57	56	expd 421	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑤 → (𝑣 ∈ ω → ((𝑤 ∖ {𝑧}) ≈ 𝑣 → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑))))
58	57	com12 33	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ ω → (𝑧 ∈ 𝑤 → ((𝑤 ∖ {𝑧}) ≈ 𝑣 → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑))))
59	58	rexlimdv 3161	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ ω → (∃𝑧 ∈ 𝑤 (𝑤 ∖ {𝑧}) ≈ 𝑣 → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑)))
60	59	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣) → (∃𝑧 ∈ 𝑤 (𝑤 ∖ {𝑧}) ≈ 𝑣 → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑)))
61	20, 60	mpd 16	. . . . . . . . . 10 ⊢ ((𝑣 ∈ ω ∧ 𝑤 ≈ suc 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑))
62	61	ex 418	. . . . . . . . 9 ⊢ (𝑣 ∈ ω → (𝑤 ≈ suc 𝑣 → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [𝑤 / 𝑥]𝜑)))
63	62	com23 87	. . . . . . . 8 ⊢ (𝑣 ∈ ω → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → (𝑤 ≈ suc 𝑣 → [𝑤 / 𝑥]𝜑)))
64	63	alrimdv 1962	. . . . . . 7 ⊢ (𝑣 ∈ ω → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → ∀𝑤(𝑤 ≈ suc 𝑣 → [𝑤 / 𝑥]𝜑)))
65		nfv 1947	. . . . . . . 8 ⊢ Ⅎ𝑤(𝑥 ≈ suc 𝑣 → 𝜑)
66		nfv 1947	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑤 ≈ suc 𝑣
67		nfsbc1v 3759	. . . . . . . . 9 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝜑
68	66, 67	nfim 1929	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑤 ≈ suc 𝑣 → [𝑤 / 𝑥]𝜑)
69		breq1 5106	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑥 ≈ suc 𝑣 ↔ 𝑤 ≈ suc 𝑣))
70		sbceq1a 3750	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑥]𝜑))
71	69, 70	imbi12d 347	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → ((𝑥 ≈ suc 𝑣 → 𝜑) ↔ (𝑤 ≈ suc 𝑣 → [𝑤 / 𝑥]𝜑)))
72	65, 68, 71	cbvalv1 2370	. . . . . . 7 ⊢ (∀𝑥(𝑥 ≈ suc 𝑣 → 𝜑) ↔ ∀𝑤(𝑤 ≈ suc 𝑣 → [𝑤 / 𝑥]𝜑))
73	64, 72	imbitrrdi 255	. . . . . 6 ⊢ (𝑣 ∈ ω → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → ∀𝑥(𝑥 ≈ suc 𝑣 → 𝜑)))
74	5, 8, 11, 17, 73	finds1 7896	. . . . 5 ⊢ (𝑤 ∈ ω → ∀𝑥(𝑥 ≈ 𝑤 → 𝜑))
75	74	19.21bi 2225	. . . 4 ⊢ (𝑤 ∈ ω → (𝑥 ≈ 𝑤 → 𝜑))
76	75	rexlimiv 3156	. . 3 ⊢ (∃𝑤 ∈ ω 𝑥 ≈ 𝑤 → 𝜑)
77	2, 76	sylbi 220	. 2 ⊢ (𝑥 ∈ Fin → 𝜑)
78	1, 77	vtoclga 3536	1 ⊢ (𝐴 ∈ Fin → 𝜏)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ∃wrex 3086  [wsbc 3739   ∖ cdif 3896   ∪ cun 3897   ⊆ wss 3899  ∅c0 4279  {csn 4584   class class class wbr 5103  Oncon0 6357  suc csuc 6359  ωcom 7862   ≈ cen 8949  Fincfn 8952

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-om 7863  df-en 8953  df-fin 8956

This theorem is used by:  findcard2s  9160  ssfi  9167  cnvfi  9170  fnfi  9172  frfi  9255  pwfi  9288  iunfi  9310  finsschain  9326  infdiffi  9637  fin1a2lem10  10411  wunfi  10730  rexfiuz  15435  modfsummod  15881  lcmfunsnlem  16731  lcmfun  16735  drsdirfi  18393  fiuncmp  23629  finiunmbl  25772  fineqvac  35642  mbfresfi  38415  heibor1lem  38559  pclfinclN  40823