Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj981 Structured version   Visualization version   GIF version

Theorem bnj981 30728
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
bnj981.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj981.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj981.3 𝐷 = (ω ∖ {∅})
bnj981.4 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
bnj981.5 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
Assertion
Ref Expression
bnj981 (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓𝑛𝑖(𝜒𝑖𝑛𝑍 ∈ (𝑓𝑖)))
Distinct variable groups:   𝐴,𝑓,𝑖,𝑛,𝑦   𝐷,𝑖,𝑦   𝑅,𝑓,𝑖,𝑛,𝑦   𝑓,𝑋,𝑖,𝑛,𝑦   𝑓,𝑍,𝑖,𝑛,𝑦   𝜑,𝑖,𝑦
Allowed substitution hints:   𝜑(𝑓,𝑛)   𝜓(𝑦,𝑓,𝑖,𝑛)   𝜒(𝑦,𝑓,𝑖,𝑛)   𝐵(𝑦,𝑓,𝑖,𝑛)   𝐷(𝑓,𝑛)

Proof of Theorem bnj981
StepHypRef Expression
1 nfv 1840 . . . 4 𝑦 𝑍 ∈ trCl(𝑋, 𝐴, 𝑅)
2 bnj981.2 . . . . . . . . . . . 12 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
3 nfcv 2761 . . . . . . . . . . . . 13 𝑦ω
4 nfv 1840 . . . . . . . . . . . . . 14 𝑦 suc 𝑖𝑛
5 nfiu1 4516 . . . . . . . . . . . . . . 15 𝑦 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)
65nfeq2 2776 . . . . . . . . . . . . . 14 𝑦(𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)
74, 6nfim 1822 . . . . . . . . . . . . 13 𝑦(suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
83, 7nfral 2940 . . . . . . . . . . . 12 𝑦𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
92, 8nfxfr 1776 . . . . . . . . . . 11 𝑦𝜓
109nf5ri 2063 . . . . . . . . . 10 (𝜓 → ∀𝑦𝜓)
11 bnj981.5 . . . . . . . . . 10 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
1210, 11bnj1096 30561 . . . . . . . . 9 (𝜒 → ∀𝑦𝜒)
1312nf5i 2021 . . . . . . . 8 𝑦𝜒
14 nfv 1840 . . . . . . . 8 𝑦 𝑖𝑛
15 nfv 1840 . . . . . . . 8 𝑦 𝑍 ∈ (𝑓𝑖)
1613, 14, 15nf3an 1828 . . . . . . 7 𝑦(𝜒𝑖𝑛𝑍 ∈ (𝑓𝑖))
1716nfex 2151 . . . . . 6 𝑦𝑖(𝜒𝑖𝑛𝑍 ∈ (𝑓𝑖))
1817nfex 2151 . . . . 5 𝑦𝑛𝑖(𝜒𝑖𝑛𝑍 ∈ (𝑓𝑖))
1918nfex 2151 . . . 4 𝑦𝑓𝑛𝑖(𝜒𝑖𝑛𝑍 ∈ (𝑓𝑖))
201, 19nfim 1822 . . 3 𝑦(𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓𝑛𝑖(𝜒𝑖𝑛𝑍 ∈ (𝑓𝑖)))
21 eleq1 2686 . . . 4 (𝑦 = 𝑍 → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ 𝑍 ∈ trCl(𝑋, 𝐴, 𝑅)))
22 eleq1 2686 . . . . . 6 (𝑦 = 𝑍 → (𝑦 ∈ (𝑓𝑖) ↔ 𝑍 ∈ (𝑓𝑖)))
23223anbi3d 1402 . . . . 5 (𝑦 = 𝑍 → ((𝜒𝑖𝑛𝑦 ∈ (𝑓𝑖)) ↔ (𝜒𝑖𝑛𝑍 ∈ (𝑓𝑖))))
24233exbidv 1850 . . . 4 (𝑦 = 𝑍 → (∃𝑓𝑛𝑖(𝜒𝑖𝑛𝑦 ∈ (𝑓𝑖)) ↔ ∃𝑓𝑛𝑖(𝜒𝑖𝑛𝑍 ∈ (𝑓𝑖))))
2521, 24imbi12d 334 . . 3 (𝑦 = 𝑍 → ((𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓𝑛𝑖(𝜒𝑖𝑛𝑦 ∈ (𝑓𝑖))) ↔ (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓𝑛𝑖(𝜒𝑖𝑛𝑍 ∈ (𝑓𝑖)))))
26 bnj981.1 . . . 4 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
27 bnj981.3 . . . 4 𝐷 = (ω ∖ {∅})
28 bnj981.4 . . . 4 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
2926, 2, 27, 28, 11bnj917 30712 . . 3 (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓𝑛𝑖(𝜒𝑖𝑛𝑦 ∈ (𝑓𝑖)))
3020, 25, 29vtoclg1f 3251 . 2 (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) → (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓𝑛𝑖(𝜒𝑖𝑛𝑍 ∈ (𝑓𝑖))))
3130pm2.43i 52 1 (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓𝑛𝑖(𝜒𝑖𝑛𝑍 ∈ (𝑓𝑖)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1036   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wral 2907  wrex 2908  cdif 3552  c0 3891  {csn 4148   ciun 4485  suc csuc 5684   Fn wfn 5842  cfv 5847  ωcom 7012  w-bnj17 30459   predc-bnj14 30461   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-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
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-v 3188  df-iun 4487  df-fn 5850  df-bnj17 30460  df-bnj18 30468
This theorem is referenced by:  bnj1128  30766
  Copyright terms: Public domain W3C validator