Theorem bnj978 32929

Description: Technical lemma for bnj69 32990. 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.)

Hypotheses

Ref	Expression
bnj978.1	⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
bnj978.2	⊢ (𝜃 → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))

Assertion

Ref	Expression
bnj978	⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → TrFo( trCl(𝑋, 𝐴, 𝑅), 𝐴, 𝑅))

Distinct variable groups:   𝑦,𝐴,𝑧   𝑦,𝑅,𝑧   𝑦,𝑋,𝑧

Allowed substitution hints:   𝜃(𝑦,𝑧)

Proof of Theorem bnj978

Step	Hyp	Ref	Expression
1		bnj978.1	. . . . . 6 ⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
2		bnj978.2	. . . . . 6 ⊢ (𝜃 → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))
3	1, 2	sylbir 234	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))
4	3	gen2 1799	. . . 4 ⊢ ∀𝑦∀𝑧((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))
5		bnj253 32683	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
6	5	imbi1i 350	. . . . . 6 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ↔ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))
7	6	2albii 1823	. . . . 5 ⊢ (∀𝑦∀𝑧((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ↔ ∀𝑦∀𝑧(((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))
8		3impexp 1357	. . . . . 6 ⊢ ((((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → (𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))))
9	8	2albii 1823	. . . . 5 ⊢ (∀𝑦∀𝑧(((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ↔ ∀𝑦∀𝑧((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → (𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))))
10		19.21v 1942	. . . . . . . 8 ⊢ (∀𝑧((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → (𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑧(𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → (𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))))
11		19.21v 1942	. . . . . . . . 9 ⊢ (∀𝑧(𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → (𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))) ↔ (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))))
12	11	imbi2i 336	. . . . . . . 8 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑧(𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → (𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))))
13	10, 12	bitri 274	. . . . . . 7 ⊢ (∀𝑧((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → (𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))))
14	13	albii 1822	. . . . . 6 ⊢ (∀𝑦∀𝑧((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → (𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))) ↔ ∀𝑦((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))))
15		19.21v 1942	. . . . . 6 ⊢ (∀𝑦((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑦(𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))))
16		df-ral 3069	. . . . . . . 8 ⊢ (∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅)∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ↔ ∀𝑦(𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))))
17	16	bicomi 223	. . . . . . 7 ⊢ (∀𝑦(𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))) ↔ ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅)∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))
18	17	imbi2i 336	. . . . . 6 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑦(𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅)∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))))
19	14, 15, 18	3bitri 297	. . . . 5 ⊢ (∀𝑦∀𝑧((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → (𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅)∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))))
20	7, 9, 19	3bitri 297	. . . 4 ⊢ (∀𝑦∀𝑧((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅)∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))))
21	4, 20	mpbi 229	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅)∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))
22		dfss2 3907	. . . 4 ⊢ ( pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅) ↔ ∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))
23	22	ralbii 3092	. . 3 ⊢ (∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅) ↔ ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅)∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))
24	21, 23	sylibr 233	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
25		df-bnj19 32676	. 2 ⊢ ( TrFo( trCl(𝑋, 𝐴, 𝑅), 𝐴, 𝑅) ↔ ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
26	24, 25	sylibr 233	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → TrFo( trCl(𝑋, 𝐴, 𝑅), 𝐴, 𝑅))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086  ∀wal 1537   ∈ wcel 2106  ∀wral 3064   ⊆ wss 3887   ∧ w-bnj17 32665   predc-bnj14 32667   FrSe w-bnj15 32671   trClc-bnj18 32673   TrFow-bnj19 32675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1088  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-v 3434  df-in 3894  df-ss 3904  df-bnj17 32666  df-bnj19 32676

This theorem is referenced by:  bnj907  32947