Theorem bnj222 35180

Description: Technical lemma for bnj229 35181. 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 6416	. . . . 5 ⊢ (𝑖 = 𝑚 → suc 𝑖 = suc 𝑚)
3	2	eleq1d 2849	. . . 4 ⊢ (𝑖 = 𝑚 → (suc 𝑖 ∈ 𝑁 ↔ suc 𝑚 ∈ 𝑁))
4	2	fveq2d 6873	. . . . 5 ⊢ (𝑖 = 𝑚 → (𝐹‘suc 𝑖) = (𝐹‘suc 𝑚))
5		fveq2 6869	. . . . . 6 ⊢ (𝑖 = 𝑚 → (𝐹‘𝑖) = (𝐹‘𝑚))
6	5	bnj1113 35083	. . . . 5 ⊢ (𝑖 = 𝑚 → ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅))
7	4, 6	eqeq12d 2780	. . . 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 346	. . 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 3242	. 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 277	1 ⊢ (𝜓 ↔ ∀𝑚 ∈ ω (suc 𝑚 ∈ 𝑁 → (𝐹‘suc 𝑚) = ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅)))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1562   ∈ wcel 2144  ∀wral 3078  ∪ ciun 4951  suc csuc 6350  ‘cfv 6523  ωcom 7848   predc-bnj14 34986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-suc 6354  df-iota 6479  df-fv 6531

This theorem is referenced by:  bnj229  35181  bnj589  35206