Theorem bnj222 35513

Description: Technical lemma for bnj229 35514. 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 6431	. . . . 5 ⊢ (𝑖 = 𝑚 → suc 𝑖 = suc 𝑚)
3	2	eleq1d 2845	. . . 4 ⊢ (𝑖 = 𝑚 → (suc 𝑖 ∈ 𝑁 ↔ suc 𝑚 ∈ 𝑁))
4	2	fveq2d 6889	. . . . 5 ⊢ (𝑖 = 𝑚 → (𝐹‘suc 𝑖) = (𝐹‘suc 𝑚))
5		fveq2 6885	. . . . . 6 ⊢ (𝑖 = 𝑚 → (𝐹‘𝑖) = (𝐹‘𝑚))
6	5	bnj1113 35416	. . . . 5 ⊢ (𝑖 = 𝑚 → ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝐹‘𝑚) pred(𝑦, 𝐴, 𝑅))
7	4, 6	eqeq12d 2776	. . . 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 3240	. 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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∪ ciun 4950  suc csuc 6364  ‘cfv 6538  ωcom 7877   predc-bnj14 35319

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-suc 6368  df-iota 6494  df-fv 6546

This theorem is used by:  bnj229  35514  bnj589  35539