Theorem bnj1112 34960

Description: Technical lemma for bnj69 34987. 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 34702	. 2 ⊢ (𝜓 ↔ ∀𝑖((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
3		eleq1w 2817	. . . . 5 ⊢ (𝑖 = 𝑗 → (𝑖 ∈ ω ↔ 𝑗 ∈ ω))
4		suceq 6419	. . . . . 6 ⊢ (𝑖 = 𝑗 → suc 𝑖 = suc 𝑗)
5	4	eleq1d 2819	. . . . 5 ⊢ (𝑖 = 𝑗 → (suc 𝑖 ∈ 𝑛 ↔ suc 𝑗 ∈ 𝑛))
6	3, 5	anbi12d 632	. . . 4 ⊢ (𝑖 = 𝑗 → ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ↔ (𝑗 ∈ ω ∧ suc 𝑗 ∈ 𝑛)))
7	4	fveq2d 6879	. . . . 5 ⊢ (𝑖 = 𝑗 → (𝑓‘suc 𝑖) = (𝑓‘suc 𝑗))
8		fveq2 6875	. . . . . 6 ⊢ (𝑖 = 𝑗 → (𝑓‘𝑖) = (𝑓‘𝑗))
9	8	bnj1113 34762	. . . . 5 ⊢ (𝑖 = 𝑗 → ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))
10	7, 9	eqeq12d 2751	. . . 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 344	. . 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	cbvalvw 2035	. 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 275	1 ⊢ (𝜓 ↔ ∀𝑗((𝑗 ∈ ω ∧ suc 𝑗 ∈ 𝑛) → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ∪ ciun 4967  suc csuc 6354  ‘cfv 6530  ωcom 7859   predc-bnj14 34665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-suc 6358  df-iota 6483  df-fv 6538

This theorem is referenced by:  bnj1118  34961