MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  findcard2s Structured version   Visualization version   GIF version

Theorem findcard2s 9205
Description: Variation of findcard2 9204 requiring that the element added in the induction step not be a member of the original set. (Contributed by Paul Chapman, 30-Nov-2012.)
Hypotheses
Ref Expression
findcard2s.1 (𝑥 = ∅ → (𝜑𝜓))
findcard2s.2 (𝑥 = 𝑦 → (𝜑𝜒))
findcard2s.3 (𝑥 = (𝑦 ∪ {𝑧}) → (𝜑𝜃))
findcard2s.4 (𝑥 = 𝐴 → (𝜑𝜏))
findcard2s.5 𝜓
findcard2s.6 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (𝜒𝜃))
Assertion
Ref Expression
findcard2s (𝐴 ∈ Fin → 𝜏)
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝜒,𝑥   𝜑,𝑦,𝑧   𝜓,𝑥   𝜏,𝑥   𝜃,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦,𝑧)   𝜒(𝑦,𝑧)   𝜃(𝑦,𝑧)   𝜏(𝑦,𝑧)

Proof of Theorem findcard2s
StepHypRef Expression
1 findcard2s.1 . 2 (𝑥 = ∅ → (𝜑𝜓))
2 findcard2s.2 . 2 (𝑥 = 𝑦 → (𝜑𝜒))
3 findcard2s.3 . 2 (𝑥 = (𝑦 ∪ {𝑧}) → (𝜑𝜃))
4 findcard2s.4 . 2 (𝑥 = 𝐴 → (𝜑𝜏))
5 findcard2s.5 . 2 𝜓
6 findcard2s.6 . . . 4 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (𝜒𝜃))
76ex 412 . . 3 (𝑦 ∈ Fin → (¬ 𝑧𝑦 → (𝜒𝜃)))
8 snssi 4808 . . . . . . . 8 (𝑧𝑦 → {𝑧} ⊆ 𝑦)
9 ssequn1 4186 . . . . . . . 8 ({𝑧} ⊆ 𝑦 ↔ ({𝑧} ∪ 𝑦) = 𝑦)
108, 9sylib 218 . . . . . . 7 (𝑧𝑦 → ({𝑧} ∪ 𝑦) = 𝑦)
11 uncom 4158 . . . . . . 7 ({𝑧} ∪ 𝑦) = (𝑦 ∪ {𝑧})
1210, 11eqtr3di 2792 . . . . . 6 (𝑧𝑦𝑦 = (𝑦 ∪ {𝑧}))
13 vex 3484 . . . . . . 7 𝑦 ∈ V
1413eqvinc 3649 . . . . . 6 (𝑦 = (𝑦 ∪ {𝑧}) ↔ ∃𝑥(𝑥 = 𝑦𝑥 = (𝑦 ∪ {𝑧})))
1512, 14sylib 218 . . . . 5 (𝑧𝑦 → ∃𝑥(𝑥 = 𝑦𝑥 = (𝑦 ∪ {𝑧})))
162bicomd 223 . . . . . . 7 (𝑥 = 𝑦 → (𝜒𝜑))
1716, 3sylan9bb 509 . . . . . 6 ((𝑥 = 𝑦𝑥 = (𝑦 ∪ {𝑧})) → (𝜒𝜃))
1817exlimiv 1930 . . . . 5 (∃𝑥(𝑥 = 𝑦𝑥 = (𝑦 ∪ {𝑧})) → (𝜒𝜃))
1915, 18syl 17 . . . 4 (𝑧𝑦 → (𝜒𝜃))
2019biimpd 229 . . 3 (𝑧𝑦 → (𝜒𝜃))
217, 20pm2.61d2 181 . 2 (𝑦 ∈ Fin → (𝜒𝜃))
221, 2, 3, 4, 5, 21findcard2 9204 1 (𝐴 ∈ Fin → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  cun 3949  wss 3951  c0 4333  {csn 4626  Fincfn 8985
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-en 8986  df-fin 8989
This theorem is referenced by:  findcard2d  9206  unfi  9211  ac6sfi  9320  fodomfi  9350  domunfican  9361  fodomfiOLD  9370  hashxplem  14472  hashmap  14474  hashbc  14492  hashf1lem2  14495  hashf1  14496  fsum2d  15807  fsumabs  15837  fsumrlim  15847  fsumo1  15848  fsumiun  15857  incexclem  15872  fprod2d  16017  coprmprod  16698  coprmproddvds  16700  gsum2dlem2  19989  ablfac1eulem  20092  mplcoe1  22055  mplcoe5  22058  coe1fzgsumd  22308  evl1gsumd  22361  mdetunilem9  22626  ptcmpfi  23821  tmdgsum  24103  fsumcn  24894  ovolfiniun  25536  volfiniun  25582  itgfsum  25862  dvmptfsum  26013  jensen  27032  gsumle  33101  gsumvsca1  33232  gsumvsca2  33233  finixpnum  37612  matunitlindflem1  37623  pwslnm  43106  fnchoice  45034  dvmptfprod  45960
  Copyright terms: Public domain W3C validator