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Theorem bnj917 30765
Description: Technical lemma for bnj69 30839. 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
bnj917.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj917.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj917.3 𝐷 = (ω ∖ {∅})
bnj917.4 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
bnj917.5 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
Assertion
Ref Expression
bnj917 (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓𝑛𝑖(𝜒𝑖𝑛𝑦 ∈ (𝑓𝑖)))
Distinct variable groups:   𝐴,𝑓,𝑖,𝑛,𝑦   𝐷,𝑖   𝑅,𝑓,𝑖,𝑛,𝑦   𝑓,𝑋,𝑖,𝑛,𝑦   𝜑,𝑖
Allowed substitution hints:   𝜑(𝑦,𝑓,𝑛)   𝜓(𝑦,𝑓,𝑖,𝑛)   𝜒(𝑦,𝑓,𝑖,𝑛)   𝐵(𝑦,𝑓,𝑖,𝑛)   𝐷(𝑦,𝑓,𝑛)

Proof of Theorem bnj917
StepHypRef Expression
1 bnj917.1 . . 3 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2 bnj917.2 . . 3 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
3 bnj917.3 . . 3 𝐷 = (ω ∖ {∅})
4 bnj917.4 . . 3 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
5 biid 251 . . 3 ((𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑓 Fn 𝑛𝜑𝜓))
61, 2, 3, 4, 5bnj916 30764 . 2 (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓𝑛𝑖(𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓) ∧ 𝑖𝑛𝑦 ∈ (𝑓𝑖)))
7 bnj917.5 . . . . . 6 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
8 bnj252 30529 . . . . . 6 ((𝑛𝐷𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
97, 8bitri 264 . . . . 5 (𝜒 ↔ (𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
1093anbi1i 1251 . . . 4 ((𝜒𝑖𝑛𝑦 ∈ (𝑓𝑖)) ↔ ((𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)) ∧ 𝑖𝑛𝑦 ∈ (𝑓𝑖)))
11 bnj253 30530 . . . 4 ((𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓) ∧ 𝑖𝑛𝑦 ∈ (𝑓𝑖)) ↔ ((𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)) ∧ 𝑖𝑛𝑦 ∈ (𝑓𝑖)))
1210, 11bitr4i 267 . . 3 ((𝜒𝑖𝑛𝑦 ∈ (𝑓𝑖)) ↔ (𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓) ∧ 𝑖𝑛𝑦 ∈ (𝑓𝑖)))
13123exbii 1773 . 2 (∃𝑓𝑛𝑖(𝜒𝑖𝑛𝑦 ∈ (𝑓𝑖)) ↔ ∃𝑓𝑛𝑖(𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓) ∧ 𝑖𝑛𝑦 ∈ (𝑓𝑖)))
146, 13sylibr 224 1 (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓𝑛𝑖(𝜒𝑖𝑛𝑦 ∈ (𝑓𝑖)))
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 2908  wrex 2909  cdif 3557  c0 3897  {csn 4155   ciun 4492  suc csuc 5694   Fn wfn 5852  cfv 5857  ωcom 7027  w-bnj17 30512   predc-bnj14 30514   trClc-bnj18 30520
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 2913  df-rex 2914  df-v 3192  df-iun 4494  df-fn 5860  df-bnj17 30513  df-bnj18 30521
This theorem is referenced by:  bnj981  30781  bnj996  30786
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