Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1021 Structured version   Visualization version   GIF version

Theorem bnj1021 32298
Description: Technical lemma for bnj69 32342. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1021.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj1021.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1021.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj1021.4 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
bnj1021.5 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
bnj1021.6 (𝜂 ↔ (𝑖𝑛𝑦 ∈ (𝑓𝑖)))
bnj1021.13 𝐷 = (ω ∖ {∅})
bnj1021.14 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
Assertion
Ref Expression
bnj1021 𝑓𝑛𝑖𝑚(𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏))
Distinct variable groups:   𝐴,𝑓,𝑖,𝑛,𝑦   𝐷,𝑖   𝑅,𝑓,𝑖,𝑛,𝑦   𝑓,𝑋,𝑖,𝑛,𝑦   𝜒,𝑚,𝑝   𝜂,𝑚,𝑝   𝜃,𝑓,𝑖,𝑛   𝜑,𝑖   𝑚,𝑛,𝜃,𝑝
Allowed substitution hints:   𝜑(𝑦,𝑧,𝑓,𝑚,𝑛,𝑝)   𝜓(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒(𝑦,𝑧,𝑓,𝑖,𝑛)   𝜃(𝑦,𝑧)   𝜏(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜂(𝑦,𝑧,𝑓,𝑖,𝑛)   𝐴(𝑧,𝑚,𝑝)   𝐵(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑦,𝑧,𝑓,𝑚,𝑛,𝑝)   𝑅(𝑧,𝑚,𝑝)   𝑋(𝑧,𝑚,𝑝)

Proof of Theorem bnj1021
StepHypRef Expression
1 bnj1021.1 . . . 4 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2 bnj1021.2 . . . 4 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
3 bnj1021.3 . . . 4 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
4 bnj1021.4 . . . 4 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
5 bnj1021.5 . . . 4 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
6 bnj1021.6 . . . 4 (𝜂 ↔ (𝑖𝑛𝑦 ∈ (𝑓𝑖)))
7 bnj1021.13 . . . 4 𝐷 = (ω ∖ {∅})
8 bnj1021.14 . . . 4 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
91, 2, 3, 4, 5, 6, 7, 8bnj996 32288 . . 3 𝑓𝑛𝑖𝑚𝑝(𝜃 → (𝜒𝜏𝜂))
10 anclb 549 . . . . . 6 ((𝜃 → (𝜒𝜏𝜂)) ↔ (𝜃 → (𝜃 ∧ (𝜒𝜏𝜂))))
11 bnj252 32033 . . . . . . 7 ((𝜃𝜒𝜏𝜂) ↔ (𝜃 ∧ (𝜒𝜏𝜂)))
1211imbi2i 339 . . . . . 6 ((𝜃 → (𝜃𝜒𝜏𝜂)) ↔ (𝜃 → (𝜃 ∧ (𝜒𝜏𝜂))))
1310, 12bitr4i 281 . . . . 5 ((𝜃 → (𝜒𝜏𝜂)) ↔ (𝜃 → (𝜃𝜒𝜏𝜂)))
14132exbii 1850 . . . 4 (∃𝑚𝑝(𝜃 → (𝜒𝜏𝜂)) ↔ ∃𝑚𝑝(𝜃 → (𝜃𝜒𝜏𝜂)))
15143exbii 1851 . . 3 (∃𝑓𝑛𝑖𝑚𝑝(𝜃 → (𝜒𝜏𝜂)) ↔ ∃𝑓𝑛𝑖𝑚𝑝(𝜃 → (𝜃𝜒𝜏𝜂)))
169, 15mpbi 233 . 2 𝑓𝑛𝑖𝑚𝑝(𝜃 → (𝜃𝜒𝜏𝜂))
17 19.37v 1999 . . . . 5 (∃𝑝(𝜃 → (𝜃𝜒𝜏𝜂)) ↔ (𝜃 → ∃𝑝(𝜃𝜒𝜏𝜂)))
18 bnj1019 32111 . . . . . 6 (∃𝑝(𝜃𝜒𝜏𝜂) ↔ (𝜃𝜒𝜂 ∧ ∃𝑝𝜏))
1918imbi2i 339 . . . . 5 ((𝜃 → ∃𝑝(𝜃𝜒𝜏𝜂)) ↔ (𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏)))
2017, 19bitri 278 . . . 4 (∃𝑝(𝜃 → (𝜃𝜒𝜏𝜂)) ↔ (𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏)))
21202exbii 1850 . . 3 (∃𝑖𝑚𝑝(𝜃 → (𝜃𝜒𝜏𝜂)) ↔ ∃𝑖𝑚(𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏)))
22212exbii 1850 . 2 (∃𝑓𝑛𝑖𝑚𝑝(𝜃 → (𝜃𝜒𝜏𝜂)) ↔ ∃𝑓𝑛𝑖𝑚(𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏)))
2316, 22mpbi 233 1 𝑓𝑛𝑖𝑚(𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wex 1781  wcel 2115  {cab 2802  wral 3133  wrex 3134  cdif 3916  c0 4276  {csn 4550   ciun 4905  suc csuc 6180   Fn wfn 6338  cfv 6343  ωcom 7574  w-bnj17 32016   predc-bnj14 32018   FrSe w-bnj15 32022   trClc-bnj18 32024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-tr 5159  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-fn 6346  df-om 7575  df-bnj17 32017  df-bnj18 32025
This theorem is referenced by:  bnj907  32299
  Copyright terms: Public domain W3C validator