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Theorem bnj917 31342
Description: Technical lemma for bnj69 31416. 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 31341 . 2 (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓𝑛𝑖(𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓) ∧ 𝑖𝑛𝑦 ∈ (𝑓𝑖)))
7 bnj917.5 . . . . . 6 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
8 bnj252 31109 . . . . . 6 ((𝑛𝐷𝑓 Fn 𝑛𝜑𝜓) ↔ (𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
97, 8bitri 264 . . . . 5 (𝜒 ↔ (𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)))
1093anbi1i 1160 . . . 4 ((𝜒𝑖𝑛𝑦 ∈ (𝑓𝑖)) ↔ ((𝑛𝐷 ∧ (𝑓 Fn 𝑛𝜑𝜓)) ∧ 𝑖𝑛𝑦 ∈ (𝑓𝑖)))
11 bnj253 31110 . . . 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 1926 . 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 382  w3a 1071   = wceq 1631  wex 1852  wcel 2145  {cab 2757  wral 3061  wrex 3062  cdif 3720  c0 4063  {csn 4317   ciun 4655  suc csuc 5867   Fn wfn 6025  cfv 6030  ωcom 7216  w-bnj17 31092   predc-bnj14 31094   trClc-bnj18 31100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-iun 4657  df-fn 6033  df-bnj17 31093  df-bnj18 31101
This theorem is referenced by:  bnj981  31358  bnj996  31363
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