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Theorem bnj590 32256
Description: Technical lemma for bnj852 32267. 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.)
Hypothesis
Ref Expression
bnj590.1 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
Assertion
Ref Expression
bnj590 ((𝐵 = suc 𝑖𝜓) → (𝑖 ∈ ω → (𝐵𝑛 → (𝑓𝐵) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))

Proof of Theorem bnj590
StepHypRef Expression
1 bnj590.1 . . . 4 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
2 rsp 3195 . . . 4 (∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) → (𝑖 ∈ ω → (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
31, 2sylbi 220 . . 3 (𝜓 → (𝑖 ∈ ω → (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
4 eleq1 2901 . . . . 5 (𝐵 = suc 𝑖 → (𝐵𝑛 ↔ suc 𝑖𝑛))
5 fveqeq2 6661 . . . . 5 (𝐵 = suc 𝑖 → ((𝑓𝐵) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
64, 5imbi12d 348 . . . 4 (𝐵 = suc 𝑖 → ((𝐵𝑛 → (𝑓𝐵) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
76imbi2d 344 . . 3 (𝐵 = suc 𝑖 → ((𝑖 ∈ ω → (𝐵𝑛 → (𝑓𝐵) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ (𝑖 ∈ ω → (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))))
83, 7syl5ibr 249 . 2 (𝐵 = suc 𝑖 → (𝜓 → (𝑖 ∈ ω → (𝐵𝑛 → (𝑓𝐵) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))))
98imp 410 1 ((𝐵 = suc 𝑖𝜓) → (𝑖 ∈ ω → (𝐵𝑛 → (𝑓𝐵) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2114  wral 3130   ciun 4894  suc csuc 6171  cfv 6334  ωcom 7565   predc-bnj14 32032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-ral 3135  df-v 3471  df-un 3913  df-in 3915  df-ss 3925  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-iota 6293  df-fv 6342
This theorem is referenced by:  bnj594  32258
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