Theorem bnj222 35237

Description: Technical lemma for bnj229 35238. 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
bnj222.1	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Assertion

Ref	Expression
bnj222	⊢ (𝜓 ↔ ∀𝑚 ∈ ω (suc 𝑚 ∈ 𝑁 → (𝐹‘suc 𝑚) = ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅)))

Distinct variable groups:   𝐴,𝑖,𝑚   𝑖,𝐹,𝑚,𝑦   𝑖,𝑁,𝑚   𝑅,𝑖,𝑚

Allowed substitution hints:   𝜓(𝑦,𝑖,𝑚)   𝐴(𝑦)   𝑅(𝑦)   𝑁(𝑦)

Proof of Theorem bnj222

Step	Hyp	Ref	Expression
1		bnj222.1	. 2 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))
2		suceq 6429	. . . . 5 ⊢ (𝑖 = 𝑚 → suc 𝑖 = suc 𝑚)
3	2	eleq1d 2846	. . . 4 ⊢ (𝑖 = 𝑚 → (suc 𝑖 ∈ 𝑁 ↔ suc 𝑚 ∈ 𝑁))
4	2	fveq2d 6885	. . . . 5 ⊢ (𝑖 = 𝑚 → (𝐹‘suc 𝑖) = (𝐹‘suc 𝑚))
5		fveq2 6881	. . . . . 6 ⊢ (𝑖 = 𝑚 → (𝐹‘𝑖) = (𝐹‘𝑚))
6	5	bnj1113 35140	. . . . 5 ⊢ (𝑖 = 𝑚 → ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅))
7	4, 6	eqeq12d 2777	. . . 4 ⊢ (𝑖 = 𝑚 → ((𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝐹‘suc 𝑚) = ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅)))
8	3, 7	imbi12d 347	. . 3 ⊢ (𝑖 = 𝑚 → ((suc 𝑖 ∈ 𝑁 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑚 ∈ 𝑁 → (𝐹‘suc 𝑚) = ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅))))
9	8	cbvralvw 3241	. 2 ⊢ (∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)) ↔ ∀𝑚 ∈ ω (suc 𝑚 ∈ 𝑁 → (𝐹‘suc 𝑚) = ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅)))
10	1, 9	bitri 278	1 ⊢ (𝜓 ↔ ∀𝑚 ∈ ω (suc 𝑚 ∈ 𝑁 → (𝐹‘suc 𝑚) = ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅)))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1568   ∈ wcel 2141  ∀wral 3077  ∪ ciun 4955  suc csuc 6362  ‘cfv 6536  ωcom 7861   predc-bnj14 35043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-suc 6366  df-iota 6492  df-fv 6544

This theorem is referenced by:  bnj229  35238  bnj589  35263