Theorem bnj1112 30305

Description: Technical lemma for bnj69 30332. 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 30045	. 2 ⊢ (𝜓 ↔ ∀𝑖((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
3		eleq1 2676	. . . . 5 ⊢ (𝑖 = 𝑗 → (𝑖 ∈ ω ↔ 𝑗 ∈ ω))
4		suceq 5707	. . . . . 6 ⊢ (𝑖 = 𝑗 → suc 𝑖 = suc 𝑗)
5	4	eleq1d 2672	. . . . 5 ⊢ (𝑖 = 𝑗 → (suc 𝑖 ∈ 𝑛 ↔ suc 𝑗 ∈ 𝑛))
6	3, 5	anbi12d 743	. . . 4 ⊢ (𝑖 = 𝑗 → ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ↔ (𝑗 ∈ ω ∧ suc 𝑗 ∈ 𝑛)))
7	4	fveq2d 6107	. . . . 5 ⊢ (𝑖 = 𝑗 → (𝑓‘suc 𝑖) = (𝑓‘suc 𝑗))
8		fveq2 6103	. . . . . 6 ⊢ (𝑖 = 𝑗 → (𝑓‘𝑖) = (𝑓‘𝑗))
9	8	bnj1113 30110	. . . . 5 ⊢ (𝑖 = 𝑗 → ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))
10	7, 9	eqeq12d 2625	. . . 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 333	. . 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	cbvalv 2261	. 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 263	1 ⊢ (𝜓 ↔ ∀𝑗((𝑗 ∈ ω ∧ suc 𝑗 ∈ 𝑛) → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∪ ciun 4455  suc csuc 5642  ‘cfv 5804  ωcom 6957   predc-bnj14 30007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-suc 5646  df-iota 5768  df-fv 5812

This theorem is referenced by:  bnj1118  30306