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Theorem bnj917 34565
Description: Technical lemma for bnj69 34641. 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 261 . . 3 ((𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑓 Fn 𝑛𝜑𝜓))
61, 2, 3, 4, 5bnj916 34564 . 2 (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓𝑛𝑖(𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓) ∧ 𝑖𝑛𝑦 ∈ (𝑓𝑖)))
7 bnj917.5 . . . . . 6 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
8 bnj252 34334 . . . . . 6 ((𝑛𝐷𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
97, 8bitri 275 . . . . 5 (𝜒 ↔ (𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
1093anbi1i 1155 . . . 4 ((𝜒𝑖𝑛𝑦 ∈ (𝑓𝑖)) ↔ ((𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)) ∧ 𝑖𝑛𝑦 ∈ (𝑓𝑖)))
11 bnj253 34335 . . . 4 ((𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓) ∧ 𝑖𝑛𝑦 ∈ (𝑓𝑖)) ↔ ((𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)) ∧ 𝑖𝑛𝑦 ∈ (𝑓𝑖)))
1210, 11bitr4i 278 . . 3 ((𝜒𝑖𝑛𝑦 ∈ (𝑓𝑖)) ↔ (𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓) ∧ 𝑖𝑛𝑦 ∈ (𝑓𝑖)))
13123exbii 1845 . 2 (∃𝑓𝑛𝑖(𝜒𝑖𝑛𝑦 ∈ (𝑓𝑖)) ↔ ∃𝑓𝑛𝑖(𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓) ∧ 𝑖𝑛𝑦 ∈ (𝑓𝑖)))
146, 13sylibr 233 1 (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓𝑛𝑖(𝜒𝑖𝑛𝑦 ∈ (𝑓𝑖)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085   = wceq 1534  wex 1774  wcel 2099  {cab 2705  wral 3058  wrex 3067  cdif 3944  c0 4323  {csn 4629   ciun 4996  suc csuc 6371   Fn wfn 6543  cfv 6548  ωcom 7870  w-bnj17 34317   predc-bnj14 34319   trClc-bnj18 34325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-v 3473  df-iun 4998  df-fn 6551  df-bnj17 34318  df-bnj18 34326
This theorem is referenced by:  bnj981  34581  bnj996  34587
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