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Theorem findcard 8353
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 8131 . . 3 (𝑥 ∈ Fin ↔ ∃𝑤 ∈ ω 𝑥𝑤)
3 breq2 4790 . . . . . . . 8 (𝑤 = ∅ → (𝑥𝑤𝑥 ≈ ∅))
43imbi1d 330 . . . . . . 7 (𝑤 = ∅ → ((𝑥𝑤𝜑) ↔ (𝑥 ≈ ∅ → 𝜑)))
54albidv 2001 . . . . . 6 (𝑤 = ∅ → (∀𝑥(𝑥𝑤𝜑) ↔ ∀𝑥(𝑥 ≈ ∅ → 𝜑)))
6 breq2 4790 . . . . . . . 8 (𝑤 = 𝑣 → (𝑥𝑤𝑥𝑣))
76imbi1d 330 . . . . . . 7 (𝑤 = 𝑣 → ((𝑥𝑤𝜑) ↔ (𝑥𝑣𝜑)))
87albidv 2001 . . . . . 6 (𝑤 = 𝑣 → (∀𝑥(𝑥𝑤𝜑) ↔ ∀𝑥(𝑥𝑣𝜑)))
9 breq2 4790 . . . . . . . 8 (𝑤 = suc 𝑣 → (𝑥𝑤𝑥 ≈ suc 𝑣))
109imbi1d 330 . . . . . . 7 (𝑤 = suc 𝑣 → ((𝑥𝑤𝜑) ↔ (𝑥 ≈ suc 𝑣𝜑)))
1110albidv 2001 . . . . . 6 (𝑤 = suc 𝑣 → (∀𝑥(𝑥𝑤𝜑) ↔ ∀𝑥(𝑥 ≈ suc 𝑣𝜑)))
12 en0 8170 . . . . . . . 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 1870 . . . . . 6 𝑥(𝑥 ≈ ∅ → 𝜑)
18 peano2 7231 . . . . . . . . . . . . 13 (𝑣 ∈ ω → suc 𝑣 ∈ ω)
19 breq2 4790 . . . . . . . . . . . . . 14 (𝑤 = suc 𝑣 → (𝑦𝑤𝑦 ≈ suc 𝑣))
2019rspcev 3460 . . . . . . . . . . . . 13 ((suc 𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) → ∃𝑤 ∈ ω 𝑦𝑤)
2118, 20sylan 569 . . . . . . . . . . . 12 ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) → ∃𝑤 ∈ ω 𝑦𝑤)
22 isfi 8131 . . . . . . . . . . . 12 (𝑦 ∈ Fin ↔ ∃𝑤 ∈ ω 𝑦𝑤)
2321, 22sylibr 224 . . . . . . . . . . 11 ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) → 𝑦 ∈ Fin)
24233adant2 1125 . . . . . . . . . 10 ((𝑣 ∈ ω ∧ ∀𝑥(𝑥𝑣𝜑) ∧ 𝑦 ≈ suc 𝑣) → 𝑦 ∈ Fin)
25 dif1en 8347 . . . . . . . . . . . . . . . 16 ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣𝑧𝑦) → (𝑦 ∖ {𝑧}) ≈ 𝑣)
26253expa 1111 . . . . . . . . . . . . . . 15 (((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) ∧ 𝑧𝑦) → (𝑦 ∖ {𝑧}) ≈ 𝑣)
27 vex 3354 . . . . . . . . . . . . . . . . 17 𝑦 ∈ V
28 difexg 4942 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ V → (𝑦 ∖ {𝑧}) ∈ V)
2927, 28ax-mp 5 . . . . . . . . . . . . . . . 16 (𝑦 ∖ {𝑧}) ∈ V
30 breq1 4789 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑦 ∖ {𝑧}) → (𝑥𝑣 ↔ (𝑦 ∖ {𝑧}) ≈ 𝑣))
31 findcard.2 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑦 ∖ {𝑧}) → (𝜑𝜒))
3230, 31imbi12d 333 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑦 ∖ {𝑧}) → ((𝑥𝑣𝜑) ↔ ((𝑦 ∖ {𝑧}) ≈ 𝑣𝜒)))
3329, 32spcv 3450 . . . . . . . . . . . . . . 15 (∀𝑥(𝑥𝑣𝜑) → ((𝑦 ∖ {𝑧}) ≈ 𝑣𝜒))
3426, 33syl5com 31 . . . . . . . . . . . . . 14 (((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) ∧ 𝑧𝑦) → (∀𝑥(𝑥𝑣𝜑) → 𝜒))
3534ralrimdva 3118 . . . . . . . . . . . . 13 ((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) → (∀𝑥(𝑥𝑣𝜑) → ∀𝑧𝑦 𝜒))
3635imp 393 . . . . . . . . . . . 12 (((𝑣 ∈ ω ∧ 𝑦 ≈ suc 𝑣) ∧ ∀𝑥(𝑥𝑣𝜑)) → ∀𝑧𝑦 𝜒)
3736an32s 631 . . . . . . . . . . 11 (((𝑣 ∈ ω ∧ ∀𝑥(𝑥𝑣𝜑)) ∧ 𝑦 ≈ suc 𝑣) → ∀𝑧𝑦 𝜒)
38373impa 1100 . . . . . . . . . 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 1112 . . . . . . . 8 (𝑣 ∈ ω → (∀𝑥(𝑥𝑣𝜑) → (𝑦 ≈ suc 𝑣𝜃)))
4241alrimdv 2009 . . . . . . 7 (𝑣 ∈ ω → (∀𝑥(𝑥𝑣𝜑) → ∀𝑦(𝑦 ≈ suc 𝑣𝜃)))
43 breq1 4789 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 ≈ suc 𝑣𝑦 ≈ suc 𝑣))
44 findcard.3 . . . . . . . . 9 (𝑥 = 𝑦 → (𝜑𝜃))
4543, 44imbi12d 333 . . . . . . . 8 (𝑥 = 𝑦 → ((𝑥 ≈ suc 𝑣𝜑) ↔ (𝑦 ≈ suc 𝑣𝜃)))
4645cbvalvw 2125 . . . . . . 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 7240 . . . . 5 (𝑤 ∈ ω → ∀𝑥(𝑥𝑤𝜑))
494819.21bi 2213 . . . 4 (𝑤 ∈ ω → (𝑥𝑤𝜑))
5049rexlimiv 3175 . . 3 (∃𝑤 ∈ ω 𝑥𝑤𝜑)
512, 50sylbi 207 . 2 (𝑥 ∈ Fin → 𝜑)
521, 51vtoclga 3423 1 (𝐴 ∈ Fin → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071  wal 1629   = wceq 1631  wcel 2145  wral 3061  wrex 3062  Vcvv 3351  cdif 3720  c0 4063  {csn 4316   class class class wbr 4786  suc csuc 5866  ωcom 7210  cen 8104  Fincfn 8107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5867  df-on 5868  df-lim 5869  df-suc 5870  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-om 7211  df-1o 7711  df-er 7894  df-en 8108  df-fin 8111
This theorem is referenced by:  xpfi  8385
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