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

Theorem findcard2s 8552
Description: Variation of findcard2 8551 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 405 . . 3 (𝑦 ∈ Fin → (¬ 𝑧𝑦 → (𝜒𝜃)))
8 uncom 4012 . . . . . . 7 ({𝑧} ∪ 𝑦) = (𝑦 ∪ {𝑧})
9 snssi 4611 . . . . . . . 8 (𝑧𝑦 → {𝑧} ⊆ 𝑦)
10 ssequn1 4038 . . . . . . . 8 ({𝑧} ⊆ 𝑦 ↔ ({𝑧} ∪ 𝑦) = 𝑦)
119, 10sylib 210 . . . . . . 7 (𝑧𝑦 → ({𝑧} ∪ 𝑦) = 𝑦)
128, 11syl5reqr 2823 . . . . . 6 (𝑧𝑦𝑦 = (𝑦 ∪ {𝑧}))
13 vex 3412 . . . . . . 7 𝑦 ∈ V
1413eqvinc 3551 . . . . . 6 (𝑦 = (𝑦 ∪ {𝑧}) ↔ ∃𝑥(𝑥 = 𝑦𝑥 = (𝑦 ∪ {𝑧})))
1512, 14sylib 210 . . . . 5 (𝑧𝑦 → ∃𝑥(𝑥 = 𝑦𝑥 = (𝑦 ∪ {𝑧})))
162bicomd 215 . . . . . . 7 (𝑥 = 𝑦 → (𝜒𝜑))
1716, 3sylan9bb 502 . . . . . 6 ((𝑥 = 𝑦𝑥 = (𝑦 ∪ {𝑧})) → (𝜒𝜃))
1817exlimiv 1889 . . . . 5 (∃𝑥(𝑥 = 𝑦𝑥 = (𝑦 ∪ {𝑧})) → (𝜒𝜃))
1915, 18syl 17 . . . 4 (𝑧𝑦 → (𝜒𝜃))
2019biimpd 221 . . 3 (𝑧𝑦 → (𝜒𝜃))
217, 20pm2.61d2 174 . 2 (𝑦 ∈ Fin → (𝜒𝜃))
221, 2, 3, 4, 5, 21findcard2 8551 1 (𝐴 ∈ Fin → 𝜏)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387   = wceq 1507  wex 1742  wcel 2050  cun 3821  wss 3823  c0 4172  {csn 4435  Fincfn 8304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3676  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-pss 3839  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-om 7395  df-1o 7903  df-er 8087  df-en 8305  df-fin 8308
This theorem is referenced by:  findcard2d  8553  ac6sfi  8555  domunfican  8584  fodomfi  8590  hashxplem  13605  hashmap  13607  hashbc  13622  hashf1lem2  13625  hashf1  13626  fsum2d  14984  fsumabs  15014  fsumrlim  15024  fsumo1  15025  fsumiun  15034  incexclem  15049  fprod2d  15193  coprmprod  15859  coprmproddvds  15861  gsum2dlem2  18856  ablfac1eulem  18956  mplcoe1  19971  mplcoe5  19974  coe1fzgsumd  20185  evl1gsumd  20234  mdetunilem9  20945  ptcmpfi  22137  tmdgsum  22419  fsumcn  23193  ovolfiniun  23817  volfiniun  23863  itgfsum  24142  dvmptfsum  24287  jensen  25280  gsumle  30551  gsumvsca1  30554  gsumvsca2  30555  finixpnum  34347  matunitlindflem1  34358  pwslnm  39119  fnchoice  40734  dvmptfprod  41685
  Copyright terms: Public domain W3C validator