Theorem findcard 7145

Description: Schema for induction on the cardinality of a finite set. The inductive hypothesis is that the result is true on the given set with any one element removed. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 2-Sep-2009.)

Hypotheses

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

Assertion

Ref	Expression
findcard	⊢ (𝐴 ∈ Fin → 𝜏)

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

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

Proof of Theorem findcard

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

Step	Hyp	Ref	Expression
1		findcard.4	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))
2		isfi 7000	. . 3 ⊢ (𝑥 ∈ Fin ↔ ∃𝑤 ∈ ω 𝑥 ≈ 𝑤)
3		breq2 4113	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ ∅))
4	3	imbi1d 231	. . . . . . 7 ⊢ (𝑤 = ∅ → ((𝑥 ≈ 𝑤 → 𝜑) ↔ (𝑥 ≈ ∅ → 𝜑)))
5	4	albidv 1873	. . . . . 6 ⊢ (𝑤 = ∅ → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ ∅ → 𝜑)))
6		breq2 4113	. . . . . . . 8 ⊢ (𝑤 = 𝑣 → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ 𝑣))
7	6	imbi1d 231	. . . . . . 7 ⊢ (𝑤 = 𝑣 → ((𝑥 ≈ 𝑤 → 𝜑) ↔ (𝑥 ≈ 𝑣 → 𝜑)))
8	7	albidv 1873	. . . . . 6 ⊢ (𝑤 = 𝑣 → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ 𝑣 → 𝜑)))
9		breq2 4113	. . . . . . . 8 ⊢ (𝑤 = suc 𝑣 → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ suc 𝑣))
10	9	imbi1d 231	. . . . . . 7 ⊢ (𝑤 = suc 𝑣 → ((𝑥 ≈ 𝑤 → 𝜑) ↔ (𝑥 ≈ suc 𝑣 → 𝜑)))
11	10	albidv 1873	. . . . . 6 ⊢ (𝑤 = suc 𝑣 → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ suc 𝑣 → 𝜑)))
12		en0 7035	. . . . . . . 8 ⊢ (𝑥 ≈ ∅ ↔ 𝑥 = ∅)
13		findcard.5	. . . . . . . . 9 ⊢ 𝜓
14		findcard.1	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))
15	13, 14	mpbiri 168	. . . . . . . 8 ⊢ (𝑥 = ∅ → 𝜑)
16	12, 15	sylbi 121	. . . . . . 7 ⊢ (𝑥 ≈ ∅ → 𝜑)
17	16	ax-gen 1498	. . . . . 6 ⊢ ∀𝑥(𝑥 ≈ ∅ → 𝜑)
18		peano2 4717	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ ω → suc 𝑣 ∈ ω)
19		breq2 4113	. . . . . . . . . . . . . 14 ⊢ (𝑤 = suc 𝑣 → (𝑦 ≈ 𝑤 ↔ 𝑦 ≈ suc 𝑣))
20	19	rspcev 2921	. . . . . . . . . . . . 13 ⊢ ((suc 𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) → ∃𝑤 ∈ ω 𝑦 ≈ 𝑤)
21	18, 20	sylan 283	. . . . . . . . . . . 12 ⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) → ∃𝑤 ∈ ω 𝑦 ≈ 𝑤)
22		isfi 7000	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ Fin ↔ ∃𝑤 ∈ ω 𝑦 ≈ 𝑤)
23	21, 22	sylibr 134	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) → 𝑦 ∈ Fin)
24	23	3adant2 1043	. . . . . . . . . 10 ⊢ ((𝑣 ∈ ω ∧ ∀𝑥(𝑥 ≈ 𝑣 → 𝜑) ∧ 𝑦 ≈ suc 𝑣) → 𝑦 ∈ Fin)
25		dif1en 7136	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣 ∧ 𝑧 ∈ 𝑦) → (𝑦 ∖ {𝑧}) ≈ 𝑣)
26	25	3expa 1230	. . . . . . . . . . . . . . 15 ⊢ (((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) ∧ 𝑧 ∈ 𝑦) → (𝑦 ∖ {𝑧}) ≈ 𝑣)
27		vex 2816	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
28		difexg 4252	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ V → (𝑦 ∖ {𝑧}) ∈ V)
29	27, 28	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∖ {𝑧}) ∈ V
30		breq1 4112	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑦 ∖ {𝑧}) → (𝑥 ≈ 𝑣 ↔ (𝑦 ∖ {𝑧}) ≈ 𝑣))
31		findcard.2	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑦 ∖ {𝑧}) → (𝜑 ↔ 𝜒))
32	30, 31	imbi12d 234	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 ∖ {𝑧}) → ((𝑥 ≈ 𝑣 → 𝜑) ↔ ((𝑦 ∖ {𝑧}) ≈ 𝑣 → 𝜒)))
33	29, 32	spcv 2911	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → ((𝑦 ∖ {𝑧}) ≈ 𝑣 → 𝜒))
34	26, 33	syl5com 29	. . . . . . . . . . . . . 14 ⊢ (((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) ∧ 𝑧 ∈ 𝑦) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → 𝜒))
35	34	ralrimdva 2622	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → ∀𝑧 ∈ 𝑦 𝜒))
36	35	imp 124	. . . . . . . . . . . 12 ⊢ (((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) ∧ ∀𝑥(𝑥 ≈ 𝑣 → 𝜑)) → ∀𝑧 ∈ 𝑦 𝜒)
37	36	an32s 570	. . . . . . . . . . 11 ⊢ (((𝑣 ∈ ω ∧ ∀𝑥(𝑥 ≈ 𝑣 → 𝜑)) ∧ 𝑦 ≈ suc 𝑣) → ∀𝑧 ∈ 𝑦 𝜒)
38	37	3impa 1221	. . . . . . . . . 10 ⊢ ((𝑣 ∈ ω ∧ ∀𝑥(𝑥 ≈ 𝑣 → 𝜑) ∧ 𝑦 ≈ suc 𝑣) → ∀𝑧 ∈ 𝑦 𝜒)
39		findcard.6	. . . . . . . . . 10 ⊢ (𝑦 ∈ Fin → (∀𝑧 ∈ 𝑦 𝜒 → 𝜃))
40	24, 38, 39	sylc 62	. . . . . . . . 9 ⊢ ((𝑣 ∈ ω ∧ ∀𝑥(𝑥 ≈ 𝑣 → 𝜑) ∧ 𝑦 ≈ suc 𝑣) → 𝜃)
41	40	3exp 1229	. . . . . . . 8 ⊢ (𝑣 ∈ ω → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → (𝑦 ≈ suc 𝑣 → 𝜃)))
42	41	alrimdv 1925	. . . . . . 7 ⊢ (𝑣 ∈ ω → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → ∀𝑦(𝑦 ≈ suc 𝑣 → 𝜃)))
43		breq1 4112	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ suc 𝑣 ↔ 𝑦 ≈ suc 𝑣))
44		findcard.3	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜃))
45	43, 44	imbi12d 234	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥 ≈ suc 𝑣 → 𝜑) ↔ (𝑦 ≈ suc 𝑣 → 𝜃)))
46	45	cbvalv 1967	. . . . . . 7 ⊢ (∀𝑥(𝑥 ≈ suc 𝑣 → 𝜑) ↔ ∀𝑦(𝑦 ≈ suc 𝑣 → 𝜃))
47	42, 46	imbitrrdi 162	. . . . . 6 ⊢ (𝑣 ∈ ω → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → ∀𝑥(𝑥 ≈ suc 𝑣 → 𝜑)))
48	5, 8, 11, 17, 47	finds1 4724	. . . . 5 ⊢ (𝑤 ∈ ω → ∀𝑥(𝑥 ≈ 𝑤 → 𝜑))
49	48	19.21bi 1607	. . . 4 ⊢ (𝑤 ∈ ω → (𝑥 ≈ 𝑤 → 𝜑))
50	49	rexlimiv 2654	. . 3 ⊢ (∃𝑤 ∈ ω 𝑥 ≈ 𝑤 → 𝜑)
51	2, 50	sylbi 121	. 2 ⊢ (𝑥 ∈ Fin → 𝜑)
52	1, 51	vtoclga 2881	1 ⊢ (𝐴 ∈ Fin → 𝜏)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005  ∀wal 1396   = wceq 1398   ∈ wcel 2203  ∀wral 2520  ∃wrex 2521  Vcvv 2813   ∖ cdif 3208  ∅c0 3508  {csn 3689   class class class wbr 4109  suc csuc 4486  ωcom 4712   ≈ cen 6973  Fincfn 6975

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-er 6767  df-en 6976  df-fin 6978

This theorem is referenced by:  xpfi  7192