Theorem bnj590 35607

Description: Technical lemma for bnj852 35618. 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 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Assertion

Ref	Expression
bnj590	⊢ ((𝐵 = suc 𝑖 ∧ 𝜓) → (𝑖 ∈ ω → (𝐵 ∈ 𝑛 → (𝑓‘𝐵) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))

Proof of Theorem bnj590

Step	Hyp	Ref	Expression
1		bnj590.1	. . . 4 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
2		rsp 3254	. . . 4 ⊢ (∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) → (𝑖 ∈ ω → (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
3	1, 2	sylbi 220	. . 3 ⊢ (𝜓 → (𝑖 ∈ ω → (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
4		eleq1 2852	. . . . 5 ⊢ (𝐵 = suc 𝑖 → (𝐵 ∈ 𝑛 ↔ suc 𝑖 ∈ 𝑛))
5		fveqeq2 6914	. . . . 5 ⊢ (𝐵 = suc 𝑖 → ((𝑓‘𝐵) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
6	4, 5	imbi12d 347	. . . 4 ⊢ (𝐵 = suc 𝑖 → ((𝐵 ∈ 𝑛 → (𝑓‘𝐵) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
7	6	imbi2d 343	. . 3 ⊢ (𝐵 = suc 𝑖 → ((𝑖 ∈ ω → (𝐵 ∈ 𝑛 → (𝑓‘𝐵) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) ↔ (𝑖 ∈ ω → (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))))
8	3, 7	imbitrrid 249	. 2 ⊢ (𝐵 = suc 𝑖 → (𝜓 → (𝑖 ∈ ω → (𝐵 ∈ 𝑛 → (𝑓‘𝐵) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))))
9	8	imp 412	1 ⊢ ((𝐵 = suc 𝑖 ∧ 𝜓) → (𝑖 ∈ ω → (𝐵 ∈ 𝑛 → (𝑓‘𝐵) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∪ ciun 4954  suc csuc 6379  ‘cfv 6555  ωcom 7898   predc-bnj14 35386

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-12 2216  ax-ext 2735

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6509  df-fv 6563

This theorem is used by:  bnj594  35609