Theorem bnj1112 35479

Description: Technical lemma for bnj69 35506. 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 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Assertion

Ref	Expression
bnj1112	⊢ (𝜓 ↔ ∀𝑗((𝑗 ∈ ω ∧ suc 𝑗 ∈ 𝑛) → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) 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

Step	Hyp	Ref	Expression
1		bnj1112.1	. . 3 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
2	1	bnj115 35222	. 2 ⊢ (𝜓 ↔ ∀𝑖((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
3		eleq1w 2845	. . . . 5 ⊢ (𝑖 = 𝑗 → (𝑖 ∈ ω ↔ 𝑗 ∈ ω))
4		suceq 6430	. . . . . 6 ⊢ (𝑖 = 𝑗 → suc 𝑖 = suc 𝑗)
5	4	eleq1d 2847	. . . . 5 ⊢ (𝑖 = 𝑗 → (suc 𝑖 ∈ 𝑛 ↔ suc 𝑗 ∈ 𝑛))
6	3, 5	anbi12d 644	. . . 4 ⊢ (𝑖 = 𝑗 → ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ↔ (𝑗 ∈ ω ∧ suc 𝑗 ∈ 𝑛)))
7	4	fveq2d 6886	. . . . 5 ⊢ (𝑖 = 𝑗 → (𝑓‘suc 𝑖) = (𝑓‘suc 𝑗))
8		fveq2 6882	. . . . . 6 ⊢ (𝑖 = 𝑗 → (𝑓‘𝑖) = (𝑓‘𝑗))
9	8	bnj1113 35282	. . . . 5 ⊢ (𝑖 = 𝑗 → ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))
10	7, 9	eqeq12d 2778	. . . 4 ⊢ (𝑖 = 𝑗 → ((𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)))
11	6, 10	imbi12d 347	. . 3 ⊢ (𝑖 = 𝑗 → (((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ((𝑗 ∈ ω ∧ suc 𝑗 ∈ 𝑛) → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))))
12	11	cbvalvw 2069	. 2 ⊢ (∀𝑖((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑗((𝑗 ∈ ω ∧ suc 𝑗 ∈ 𝑛) → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)))
13	2, 12	bitri 278	1 ⊢ (𝜓 ↔ ∀𝑗((𝑗 ∈ ω ∧ suc 𝑗 ∈ 𝑛) → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ∪ ciun 4954  suc csuc 6363  ‘cfv 6537  ωcom 7865   predc-bnj14 35185

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-suc 6367  df-iota 6493  df-fv 6545

This theorem is used by:  bnj1118  35480