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Theorem bnj1112 34997
Description: Technical lemma for bnj69 35024. 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 34739 . 2 (𝜓 ↔ ∀𝑖((𝑖 ∈ ω ∧ suc 𝑖𝑛) → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
3 eleq1w 2824 . . . . 5 (𝑖 = 𝑗 → (𝑖 ∈ ω ↔ 𝑗 ∈ ω))
4 suceq 6450 . . . . . 6 (𝑖 = 𝑗 → suc 𝑖 = suc 𝑗)
54eleq1d 2826 . . . . 5 (𝑖 = 𝑗 → (suc 𝑖𝑛 ↔ suc 𝑗𝑛))
63, 5anbi12d 632 . . . 4 (𝑖 = 𝑗 → ((𝑖 ∈ ω ∧ suc 𝑖𝑛) ↔ (𝑗 ∈ ω ∧ suc 𝑗𝑛)))
74fveq2d 6910 . . . . 5 (𝑖 = 𝑗 → (𝑓‘suc 𝑖) = (𝑓‘suc 𝑗))
8 fveq2 6906 . . . . . 6 (𝑖 = 𝑗 → (𝑓𝑖) = (𝑓𝑗))
98bnj1113 34799 . . . . 5 (𝑖 = 𝑗 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅))
107, 9eqeq12d 2753 . . . 4 (𝑖 = 𝑗 → ((𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝑓‘suc 𝑗) = 𝑦 ∈ (𝑓𝑗) pred(𝑦, 𝐴, 𝑅)))
116, 10imbi12d 344 . . 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 2035 . 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 275 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 206  wa 395  wal 1538   = wceq 1540  wcel 2108  wral 3061   ciun 4991  suc csuc 6386  cfv 6561  ωcom 7887   predc-bnj14 34702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-suc 6390  df-iota 6514  df-fv 6569
This theorem is referenced by:  bnj1118  34998
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