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Theorem bnj590 31826
 Description: Technical lemma for bnj852 31837. 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 3156 . . . 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 209 . . 3 (𝜓 → (𝑖 ∈ ω → (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
4 eleq1 2854 . . . . 5 (𝐵 = suc 𝑖 → (𝐵𝑛 ↔ suc 𝑖𝑛))
5 fveqeq2 6508 . . . . 5 (𝐵 = suc 𝑖 → ((𝑓𝐵) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
64, 5imbi12d 337 . . . 4 (𝐵 = suc 𝑖 → ((𝐵𝑛 → (𝑓𝐵) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))))
76imbi2d 333 . . 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 238 . 2 (𝐵 = suc 𝑖 → (𝜓 → (𝑖 ∈ ω → (𝐵𝑛 → (𝑓𝐵) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))))
98imp 398 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 198   ∧ wa 387   = wceq 1507   ∈ wcel 2050  ∀wral 3089  ∪ ciun 4792  suc csuc 6031  ‘cfv 6188  ωcom 7396   predc-bnj14 31603 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2751 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-iota 6152  df-fv 6196 This theorem is referenced by:  bnj594  31828
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