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Theorem bnj1112 32537
Description: Technical lemma for bnj69 32564. 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
bnj1112.1 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
Assertion
Ref Expression
bnj1112 (𝜓 ↔ ∀𝑗((𝑗 ∈ ω ∧ suc 𝑗𝑛) → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
Distinct variable groups:   𝐴,𝑖,𝑗   𝑅,𝑖,𝑗   𝑓,𝑖,𝑗,𝑦   𝑖,𝑛,𝑗
Allowed substitution hints:   𝜓(𝑦,𝑓,𝑖,𝑗,𝑛)   𝐴(𝑦,𝑓,𝑛)   𝑅(𝑦,𝑓,𝑛)

Proof of Theorem bnj1112
StepHypRef Expression
1 bnj1112.1 . . 3 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
21bnj115 32277 . 2 (𝜓 ↔ ∀𝑖((𝑖 ∈ ω ∧ suc 𝑖𝑛) → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
3 eleq1w 2816 . . . . 5 (𝑖 = 𝑗 → (𝑖 ∈ ω ↔ 𝑗 ∈ ω))
4 suceq 6238 . . . . . 6 (𝑖 = 𝑗 → suc 𝑖 = suc 𝑗)
54eleq1d 2818 . . . . 5 (𝑖 = 𝑗 → (suc 𝑖𝑛 ↔ suc 𝑗𝑛))
63, 5anbi12d 634 . . . 4 (𝑖 = 𝑗 → ((𝑖 ∈ ω ∧ suc 𝑖𝑛) ↔ (𝑗 ∈ ω ∧ suc 𝑗𝑛)))
74fveq2d 6681 . . . . 5 (𝑖 = 𝑗 → (𝑓‘suc 𝑖) = (𝑓‘suc 𝑗))
8 fveq2 6677 . . . . . 6 (𝑖 = 𝑗 → (𝑓𝑖) = (𝑓𝑗))
98bnj1113 32339 . . . . 5 (𝑖 = 𝑗 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅))
107, 9eqeq12d 2755 . . . 4 (𝑖 = 𝑗 → ((𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
116, 10imbi12d 348 . . 3 (𝑖 = 𝑗 → (((𝑖 ∈ ω ∧ suc 𝑖𝑛) → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ((𝑗 ∈ ω ∧ suc 𝑗𝑛) → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅))))
1211cbvalvw 2048 . 2 (∀𝑖((𝑖 ∈ ω ∧ suc 𝑖𝑛) → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑗((𝑗 ∈ ω ∧ suc 𝑗𝑛) → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
132, 12bitri 278 1 (𝜓 ↔ ∀𝑗((𝑗 ∈ ω ∧ suc 𝑗𝑛) → (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1540   = wceq 1542  wcel 2114  wral 3054   ciun 4882  suc csuc 6175  cfv 6340  ωcom 7602   predc-bnj14 32240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-ral 3059  df-rex 3060  df-v 3401  df-un 3849  df-in 3851  df-ss 3861  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-iun 4884  df-br 5032  df-suc 6179  df-iota 6298  df-fv 6348
This theorem is referenced by:  bnj1118  32538
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