Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1018 Structured version   Visualization version   GIF version

Theorem bnj1018 30740
Description: Technical lemma for bnj69 30786. 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
bnj1018.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj1018.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1018.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj1018.4 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
bnj1018.5 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
bnj1018.7 (𝜑′[𝑝 / 𝑛]𝜑)
bnj1018.8 (𝜓′[𝑝 / 𝑛]𝜓)
bnj1018.9 (𝜒′[𝑝 / 𝑛]𝜒)
bnj1018.10 (𝜑″[𝐺 / 𝑓]𝜑′)
bnj1018.11 (𝜓″[𝐺 / 𝑓]𝜓′)
bnj1018.12 (𝜒″[𝐺 / 𝑓]𝜒′)
bnj1018.13 𝐷 = (ω ∖ {∅})
bnj1018.14 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
bnj1018.15 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
bnj1018.16 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
bnj1018.26 (𝜒″ ↔ (𝑝𝐷𝐺 Fn 𝑝𝜑″𝜓″))
bnj1018.29 ((𝜃𝜒𝜏𝜂) → 𝜒″)
bnj1018.30 ((𝜃𝜒𝜏𝜂) → (𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝))
Assertion
Ref Expression
bnj1018 ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → (𝐺‘suc 𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
Distinct variable groups:   𝐴,𝑓,𝑖,𝑚,𝑛,𝑦   𝐷,𝑓,𝑖,𝑛   𝑖,𝐺,𝑝   𝑅,𝑓,𝑖,𝑚,𝑛,𝑦   𝑓,𝑋,𝑖,𝑛,𝑦   𝜒,𝑝   𝜂,𝑝   𝑓,𝑝,𝑛   𝜑,𝑖   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,𝑚,𝑛)   𝜃(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛)   𝜏(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜂(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛)   𝐴(𝑧,𝑝)   𝐵(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐶(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑦,𝑧,𝑚,𝑝)   𝑅(𝑧,𝑝)   𝐺(𝑦,𝑧,𝑓,𝑚,𝑛)   𝑋(𝑧,𝑚,𝑝)   𝜑′(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓′(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒′(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜑″(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓″(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒″(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj1018
StepHypRef Expression
1 df-bnj17 30460 . . 3 ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) ↔ ((𝜃𝜒𝜂) ∧ ∃𝑝𝜏))
2 bnj258 30481 . . . . . . . 8 ((𝜃𝜒𝜏𝜂) ↔ ((𝜃𝜒𝜂) ∧ 𝜏))
3 bnj1018.29 . . . . . . . 8 ((𝜃𝜒𝜏𝜂) → 𝜒″)
42, 3sylbir 225 . . . . . . 7 (((𝜃𝜒𝜂) ∧ 𝜏) → 𝜒″)
54ex 450 . . . . . 6 ((𝜃𝜒𝜂) → (𝜏𝜒″))
65eximdv 1843 . . . . 5 ((𝜃𝜒𝜂) → (∃𝑝𝜏 → ∃𝑝𝜒″))
7 bnj1018.3 . . . . . 6 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
8 bnj1018.9 . . . . . 6 (𝜒′[𝑝 / 𝑛]𝜒)
9 bnj1018.12 . . . . . 6 (𝜒″[𝐺 / 𝑓]𝜒′)
10 bnj1018.14 . . . . . 6 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
11 bnj1018.16 . . . . . 6 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
127, 8, 9, 10, 11bnj985 30731 . . . . 5 (𝐺𝐵 ↔ ∃𝑝𝜒″)
136, 12syl6ibr 242 . . . 4 ((𝜃𝜒𝜂) → (∃𝑝𝜏𝐺𝐵))
1413imp 445 . . 3 (((𝜃𝜒𝜂) ∧ ∃𝑝𝜏) → 𝐺𝐵)
151, 14sylbi 207 . 2 ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → 𝐺𝐵)
16 bnj1019 30558 . . 3 (∃𝑝(𝜃𝜒𝜏𝜂) ↔ (𝜃𝜒𝜂 ∧ ∃𝑝𝜏))
17 bnj1018.30 . . . . . 6 ((𝜃𝜒𝜏𝜂) → (𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝))
1817simp3d 1073 . . . . 5 ((𝜃𝜒𝜏𝜂) → suc 𝑖𝑝)
19 bnj1018.26 . . . . . . 7 (𝜒″ ↔ (𝑝𝐷𝐺 Fn 𝑝𝜑″𝜓″))
2019bnj1235 30583 . . . . . 6 (𝜒″𝐺 Fn 𝑝)
21 fndm 5948 . . . . . 6 (𝐺 Fn 𝑝 → dom 𝐺 = 𝑝)
223, 20, 213syl 18 . . . . 5 ((𝜃𝜒𝜏𝜂) → dom 𝐺 = 𝑝)
2318, 22eleqtrrd 2701 . . . 4 ((𝜃𝜒𝜏𝜂) → suc 𝑖 ∈ dom 𝐺)
2423exlimiv 1855 . . 3 (∃𝑝(𝜃𝜒𝜏𝜂) → suc 𝑖 ∈ dom 𝐺)
2516, 24sylbir 225 . 2 ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → suc 𝑖 ∈ dom 𝐺)
26 bnj1018.1 . . 3 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
27 bnj1018.2 . . 3 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
28 bnj1018.13 . . 3 𝐷 = (ω ∖ {∅})
2911bnj918 30544 . . 3 𝐺 ∈ V
30 vex 3189 . . . 4 𝑖 ∈ V
3130sucex 6958 . . 3 suc 𝑖 ∈ V
3226, 27, 28, 10, 29, 31bnj1015 30739 . 2 ((𝐺𝐵 ∧ suc 𝑖 ∈ dom 𝐺) → (𝐺‘suc 𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
3315, 25, 32syl2anc 692 1 ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → (𝐺‘suc 𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wral 2907  wrex 2908  Vcvv 3186  [wsbc 3417  cdif 3552  cun 3553  wss 3555  c0 3891  {csn 4148  cop 4154   ciun 4485  dom cdm 5074  suc csuc 5684   Fn wfn 5842  cfv 5847  ωcom 7012  w-bnj17 30459   predc-bnj14 30461   FrSe w-bnj15 30465   trClc-bnj18 30467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-dm 5084  df-suc 5688  df-iota 5810  df-fn 5850  df-fv 5855  df-bnj17 30460  df-bnj18 30468
This theorem is referenced by:  bnj1020  30741
  Copyright terms: Public domain W3C validator