Theorem findcard2 9128

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 5320. (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 8947	. . 3 ⊢ (𝑥 ∈ Fin ↔ ∃𝑤 ∈ ω 𝑥 ≈ 𝑤)
3		breq2 5111	. . . . . . . 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 5111	. . . . . . . 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 5111	. . . . . . . 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 8989	. . . . . . . 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 7848	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ ω → 𝑣 ∈ On)
19		rexdif1en 9122	. . . . . . . . . . . 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 4772	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑤 → {𝑧} ⊆ 𝑤)
22		uncom 4121	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∖ {𝑧}) ∪ {𝑧}) = ({𝑧} ∪ (𝑤 ∖ {𝑧}))
23		undif 4445	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑧} ⊆ 𝑤 ↔ ({𝑧} ∪ (𝑤 ∖ {𝑧})) = 𝑤)
24	23	biimpi 216	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑧} ⊆ 𝑤 → ({𝑧} ∪ (𝑤 ∖ {𝑧})) = 𝑤)
25	22, 24	eqtrid 2776	. . . . . . . . . . . . . . . 16 ⊢ ({𝑧} ⊆ 𝑤 → ((𝑤 ∖ {𝑧}) ∪ {𝑧}) = 𝑤)
26		vex 3451	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑤 ∈ V
27	26	difexi 5285	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∖ {𝑧}) ∈ V
28		breq1 5110	. . . . . . . . . . . . . . . . . . . 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 4124	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = (𝑤 ∖ {𝑧}) → (𝑦 ∪ {𝑧}) = ((𝑤 ∖ {𝑧}) ∪ {𝑧}))
31	30	sbceq1d 3758	. . . . . . . . . . . . . . . . . . . 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 5110	. . . . . . . . . . . . . . . . . . . . . 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 1988	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → (𝑦 ≈ 𝑣 → 𝜒))
38		rspe 3227	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ 𝑣) → ∃𝑣 ∈ ω 𝑦 ≈ 𝑣)
39		isfi 8947	. . . . . . . . . . . . . . . . . . . . . 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 3451	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑦 ∈ V
47		vsnex 5389	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {𝑧} ∈ V
48	46, 47	unex 7720	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∪ {𝑧}) ∈ V
49		findcard2.3	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝜑 ↔ 𝜃))
50	48, 49	sbcie 3795	. . . . . . . . . . . . . . . . . . 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 3524	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑣 ∈ ω ∧ (𝑤 ∖ {𝑧}) ≈ 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → [((𝑤 ∖ {𝑧}) ∪ {𝑧}) / 𝑥]𝜑))
53		dfsbcq 3755	. . . . . . . . . . . . . . . . . 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 3132	. . . . . . . . . . . 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 3773	. . . . . . . . 9 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝜑
68	66, 67	nfim 1896	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑤 ≈ suc 𝑣 → [𝑤 / 𝑥]𝜑)
69		breq1 5110	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑥 ≈ suc 𝑣 ↔ 𝑤 ≈ suc 𝑣))
70		sbceq1a 3764	. . . . . . . . 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 2339	. . . . . . 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 7875	. . . . 5 ⊢ (𝑤 ∈ ω → ∀𝑥(𝑥 ≈ 𝑤 → 𝜑))
75	74	19.21bi 2190	. . . 4 ⊢ (𝑤 ∈ ω → (𝑥 ≈ 𝑤 → 𝜑))
76	75	rexlimiv 3127	. . 3 ⊢ (∃𝑤 ∈ ω 𝑥 ≈ 𝑤 → 𝜑)
77	2, 76	sylbi 217	. 2 ⊢ (𝑥 ∈ Fin → 𝜑)
78	1, 77	vtoclga 3543	1 ⊢ (𝐴 ∈ Fin → 𝜏)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  [wsbc 3753   ∖ cdif 3911   ∪ cun 3912   ⊆ wss 3914  ∅c0 4296  {csn 4589   class class class wbr 5107  Oncon0 6332  suc csuc 6334  ωcom 7842   ≈ cen 8915  Fincfn 8918

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-om 7843  df-en 8919  df-fin 8922

This theorem is referenced by:  findcard2s  9129  ssfi  9137  cnvfi  9140  fnfi  9142  frfi  9232  imafiOLD  9265  pwfi  9268  iunfi  9294  finsschain  9310  infdiffi  9611  fin1a2lem10  10362  wunfi  10674  rexfiuz  15314  modfsummod  15760  lcmfunsnlem  16611  lcmfun  16615  drsdirfi  18266  fiuncmp  23291  finiunmbl  25445  fineqvac  35087  mbfresfi  37660  heibor1lem  37803  pclfinclN  39944