Theorem findcard 7182

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 7037	. . 3 ⊢ (𝑥 ∈ Fin ↔ ∃𝑤 ∈ ω 𝑥 ≈ 𝑤)
3		breq2 4129	. . . . . . . 8 ⊢ (𝑤 = ∅ → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ ∅))
4	3	imbi1d 231	. . . . . . 7 ⊢ (𝑤 = ∅ → ((𝑥 ≈ 𝑤 → 𝜑) ↔ (𝑥 ≈ ∅ → 𝜑)))
5	4	albidv 1877	. . . . . 6 ⊢ (𝑤 = ∅ → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ ∅ → 𝜑)))
6		breq2 4129	. . . . . . . 8 ⊢ (𝑤 = 𝑣 → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ 𝑣))
7	6	imbi1d 231	. . . . . . 7 ⊢ (𝑤 = 𝑣 → ((𝑥 ≈ 𝑤 → 𝜑) ↔ (𝑥 ≈ 𝑣 → 𝜑)))
8	7	albidv 1877	. . . . . 6 ⊢ (𝑤 = 𝑣 → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ 𝑣 → 𝜑)))
9		breq2 4129	. . . . . . . 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 1877	. . . . . 6 ⊢ (𝑤 = suc 𝑣 → (∀𝑥(𝑥 ≈ 𝑤 → 𝜑) ↔ ∀𝑥(𝑥 ≈ suc 𝑣 → 𝜑)))
12		en0 7072	. . . . . . . 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 1502	. . . . . 6 ⊢ ∀𝑥(𝑥 ≈ ∅ → 𝜑)
18		peano2 4737	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ ω → suc 𝑣 ∈ ω)
19		breq2 4129	. . . . . . . . . . . . . 14 ⊢ (𝑤 = suc 𝑣 → (𝑦 ≈ 𝑤 ↔ 𝑦 ≈ suc 𝑣))
20	19	rspcev 2929	. . . . . . . . . . . . 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 7037	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ Fin ↔ ∃𝑤 ∈ ω 𝑦 ≈ 𝑤)
23	21, 22	sylibr 134	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) → 𝑦 ∈ Fin)
24	23	3adant2 1047	. . . . . . . . . 10 ⊢ ((𝑣 ∈ ω ∧ ∀𝑥(𝑥 ≈ 𝑣 → 𝜑) ∧ 𝑦 ≈ suc 𝑣) → 𝑦 ∈ Fin)
25		dif1en 7173	. . . . . . . . . . . . . . . 16 ⊢ ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣 ∧ 𝑧 ∈ 𝑦) → (𝑦 ∖ {𝑧}) ≈ 𝑣)
26	25	3expa 1234	. . . . . . . . . . . . . . 15 ⊢ (((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) ∧ 𝑧 ∈ 𝑦) → (𝑦 ∖ {𝑧}) ≈ 𝑣)
27		vex 2824	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
28		difexg 4270	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ V → (𝑦 ∖ {𝑧}) ∈ V)
29	27, 28	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∖ {𝑧}) ∈ V
30		breq1 4128	. . . . . . . . . . . . . . . . 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 2919	. . . . . . . . . . . . . . 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 2630	. . . . . . . . . . . . 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 574	. . . . . . . . . . 11 ⊢ (((𝑣 ∈ ω ∧ ∀𝑥(𝑥 ≈ 𝑣 → 𝜑)) ∧ 𝑦 ≈ suc 𝑣) → ∀𝑧 ∈ 𝑦 𝜒)
38	37	3impa 1225	. . . . . . . . . 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 1233	. . . . . . . 8 ⊢ (𝑣 ∈ ω → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → (𝑦 ≈ suc 𝑣 → 𝜃)))
42	41	alrimdv 1929	. . . . . . 7 ⊢ (𝑣 ∈ ω → (∀𝑥(𝑥 ≈ 𝑣 → 𝜑) → ∀𝑦(𝑦 ≈ suc 𝑣 → 𝜃)))
43		breq1 4128	. . . . . . . . 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 1973	. . . . . . 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 4744	. . . . 5 ⊢ (𝑤 ∈ ω → ∀𝑥(𝑥 ≈ 𝑤 → 𝜑))
49	48	19.21bi 1611	. . . 4 ⊢ (𝑤 ∈ ω → (𝑥 ≈ 𝑤 → 𝜑))
50	49	rexlimiv 2662	. . 3 ⊢ (∃𝑤 ∈ ω 𝑥 ≈ 𝑤 → 𝜑)
51	2, 50	sylbi 121	. 2 ⊢ (𝑥 ∈ Fin → 𝜑)
52	1, 51	vtoclga 2889	1 ⊢ (𝐴 ∈ Fin → 𝜏)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009  ∀wal 1400   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  Vcvv 2821   ∖ cdif 3217  ∅c0 3520  {csn 3705   class class class wbr 4125  suc csuc 4505  ωcom 4732   ≈ cen 7010  Fincfn 7012

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-er 6797  df-en 7013  df-fin 7015

This theorem is referenced by:  xpfi  7229