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Theorem findcard 8184
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 = ∅ → (𝜑𝜓))
findcard.2 (𝑥 = (𝑦 ∖ {𝑧}) → (𝜑𝜒))
findcard.3 (𝑥 = 𝑦 → (𝜑𝜃))
findcard.4 (𝑥 = 𝐴 → (𝜑𝜏))
findcard.5 𝜓
findcard.6 (𝑦 ∈ Fin → (∀𝑧𝑦 𝜒𝜃))
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.
StepHypRef Expression
1 findcard.4 . 2 (𝑥 = 𝐴 → (𝜑𝜏))
2 isfi 7964 . . 3 (𝑥 ∈ Fin ↔ ∃𝑤 ∈ ω 𝑥𝑤)
3 breq2 4648 . . . . . . . 8 (𝑤 = ∅ → (𝑥𝑤𝑥 ≈ ∅))
43imbi1d 331 . . . . . . 7 (𝑤 = ∅ → ((𝑥𝑤𝜑) ↔ (𝑥 ≈ ∅ → 𝜑)))
54albidv 1847 . . . . . 6 (𝑤 = ∅ → (∀𝑥(𝑥𝑤𝜑) ↔ ∀𝑥(𝑥 ≈ ∅ → 𝜑)))
6 breq2 4648 . . . . . . . 8 (𝑤 = 𝑣 → (𝑥𝑤𝑥𝑣))
76imbi1d 331 . . . . . . 7 (𝑤 = 𝑣 → ((𝑥𝑤𝜑) ↔ (𝑥𝑣𝜑)))
87albidv 1847 . . . . . 6 (𝑤 = 𝑣 → (∀𝑥(𝑥𝑤𝜑) ↔ ∀𝑥(𝑥𝑣𝜑)))
9 breq2 4648 . . . . . . . 8 (𝑤 = suc 𝑣 → (𝑥𝑤𝑥 ≈ suc 𝑣))
109imbi1d 331 . . . . . . 7 (𝑤 = suc 𝑣 → ((𝑥𝑤𝜑) ↔ (𝑥 ≈ suc 𝑣𝜑)))
1110albidv 1847 . . . . . 6 (𝑤 = suc 𝑣 → (∀𝑥(𝑥𝑤𝜑) ↔ ∀𝑥(𝑥 ≈ suc 𝑣𝜑)))
12 en0 8004 . . . . . . . 8 (𝑥 ≈ ∅ ↔ 𝑥 = ∅)
13 findcard.5 . . . . . . . . 9 𝜓
14 findcard.1 . . . . . . . . 9 (𝑥 = ∅ → (𝜑𝜓))
1513, 14mpbiri 248 . . . . . . . 8 (𝑥 = ∅ → 𝜑)
1612, 15sylbi 207 . . . . . . 7 (𝑥 ≈ ∅ → 𝜑)
1716ax-gen 1720 . . . . . 6 𝑥(𝑥 ≈ ∅ → 𝜑)
18 peano2 7071 . . . . . . . . . . . . 13 (𝑣 ∈ ω → suc 𝑣 ∈ ω)
19 breq2 4648 . . . . . . . . . . . . . 14 (𝑤 = suc 𝑣 → (𝑦𝑤𝑦 ≈ suc 𝑣))
2019rspcev 3304 . . . . . . . . . . . . 13 ((suc 𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) → ∃𝑤 ∈ ω 𝑦𝑤)
2118, 20sylan 488 . . . . . . . . . . . 12 ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) → ∃𝑤 ∈ ω 𝑦𝑤)
22 isfi 7964 . . . . . . . . . . . 12 (𝑦 ∈ Fin ↔ ∃𝑤 ∈ ω 𝑦𝑤)
2321, 22sylibr 224 . . . . . . . . . . 11 ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) → 𝑦 ∈ Fin)
24233adant2 1078 . . . . . . . . . 10 ((𝑣 ∈ ω ∧ ∀𝑥(𝑥𝑣𝜑) ∧ 𝑦 ≈ suc 𝑣) → 𝑦 ∈ Fin)
25 dif1en 8178 . . . . . . . . . . . . . . . 16 ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣𝑧𝑦) → (𝑦 ∖ {𝑧}) ≈ 𝑣)
26253expa 1263 . . . . . . . . . . . . . . 15 (((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) ∧ 𝑧𝑦) → (𝑦 ∖ {𝑧}) ≈ 𝑣)
27 vex 3198 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
28 difexg 4799 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ V → (𝑦 ∖ {𝑧}) ∈ V)
2927, 28ax-mp 5 . . . . . . . . . . . . . . . 16 (𝑦 ∖ {𝑧}) ∈ V
30 breq1 4647 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑦 ∖ {𝑧}) → (𝑥𝑣 ↔ (𝑦 ∖ {𝑧}) ≈ 𝑣))
31 findcard.2 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑦 ∖ {𝑧}) → (𝜑𝜒))
3230, 31imbi12d 334 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑦 ∖ {𝑧}) → ((𝑥𝑣𝜑) ↔ ((𝑦 ∖ {𝑧}) ≈ 𝑣𝜒)))
3329, 32spcv 3294 . . . . . . . . . . . . . . 15 (∀𝑥(𝑥𝑣𝜑) → ((𝑦 ∖ {𝑧}) ≈ 𝑣𝜒))
3426, 33syl5com 31 . . . . . . . . . . . . . 14 (((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) ∧ 𝑧𝑦) → (∀𝑥(𝑥𝑣𝜑) → 𝜒))
3534ralrimdva 2966 . . . . . . . . . . . . 13 ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) → (∀𝑥(𝑥𝑣𝜑) → ∀𝑧𝑦 𝜒))
3635imp 445 . . . . . . . . . . . 12 (((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) ∧ ∀𝑥(𝑥𝑣𝜑)) → ∀𝑧𝑦 𝜒)
3736an32s 845 . . . . . . . . . . 11 (((𝑣 ∈ ω ∧ ∀𝑥(𝑥𝑣𝜑)) ∧ 𝑦 ≈ suc 𝑣) → ∀𝑧𝑦 𝜒)
38373impa 1257 . . . . . . . . . 10 ((𝑣 ∈ ω ∧ ∀𝑥(𝑥𝑣𝜑) ∧ 𝑦 ≈ suc 𝑣) → ∀𝑧𝑦 𝜒)
39 findcard.6 . . . . . . . . . 10 (𝑦 ∈ Fin → (∀𝑧𝑦 𝜒𝜃))
4024, 38, 39sylc 65 . . . . . . . . 9 ((𝑣 ∈ ω ∧ ∀𝑥(𝑥𝑣𝜑) ∧ 𝑦 ≈ suc 𝑣) → 𝜃)
41403exp 1262 . . . . . . . 8 (𝑣 ∈ ω → (∀𝑥(𝑥𝑣𝜑) → (𝑦 ≈ suc 𝑣𝜃)))
4241alrimdv 1855 . . . . . . 7 (𝑣 ∈ ω → (∀𝑥(𝑥𝑣𝜑) → ∀𝑦(𝑦 ≈ suc 𝑣𝜃)))
43 breq1 4647 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 ≈ suc 𝑣𝑦 ≈ suc 𝑣))
44 findcard.3 . . . . . . . . 9 (𝑥 = 𝑦 → (𝜑𝜃))
4543, 44imbi12d 334 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥 ≈ suc 𝑣𝜑) ↔ (𝑦 ≈ suc 𝑣𝜃)))
4645cbvalv 2271 . . . . . . 7 (∀𝑥(𝑥 ≈ suc 𝑣𝜑) ↔ ∀𝑦(𝑦 ≈ suc 𝑣𝜃))
4742, 46syl6ibr 242 . . . . . 6 (𝑣 ∈ ω → (∀𝑥(𝑥𝑣𝜑) → ∀𝑥(𝑥 ≈ suc 𝑣𝜑)))
485, 8, 11, 17, 47finds1 7080 . . . . 5 (𝑤 ∈ ω → ∀𝑥(𝑥𝑤𝜑))
494819.21bi 2057 . . . 4 (𝑤 ∈ ω → (𝑥𝑤𝜑))
5049rexlimiv 3023 . . 3 (∃𝑤 ∈ ω 𝑥𝑤𝜑)
512, 50sylbi 207 . 2 (𝑥 ∈ Fin → 𝜑)
521, 51vtoclga 3267 1 (𝐴 ∈ Fin → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wal 1479   = wceq 1481  wcel 1988  wral 2909  wrex 2910  Vcvv 3195  cdif 3564  c0 3907  {csn 4168   class class class wbr 4644  suc csuc 5713  ωcom 7050  cen 7937  Fincfn 7940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-om 7051  df-1o 7545  df-er 7727  df-en 7941  df-fin 7944
This theorem is referenced by:  xpfi  8216
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