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Theorem bnj978 30762
Description: Technical lemma for bnj69 30821. 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 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
bnj978.2 (𝜃𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))
Assertion
Ref Expression
bnj978 ((𝑅 FrSe 𝐴𝑋𝐴) → 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
StepHypRef Expression
1 bnj978.1 . . . . . 6 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
2 bnj978.2 . . . . . 6 (𝜃𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))
31, 2sylbir 225 . . . . 5 ((𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))
43gen2 1720 . . . 4 𝑦𝑧((𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))
5 bnj253 30512 . . . . . . 7 ((𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
65imbi1i 339 . . . . . 6 (((𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ↔ (((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))
762albii 1745 . . . . 5 (∀𝑦𝑧((𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ↔ ∀𝑦𝑧(((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))
8 3impexp 1286 . . . . . 6 ((((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → (𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))))
982albii 1745 . . . . 5 (∀𝑦𝑧(((𝑅 FrSe 𝐴𝑋𝐴) ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ↔ ∀𝑦𝑧((𝑅 FrSe 𝐴𝑋𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → (𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))))
10 19.21v 1865 . . . . . . . 8 (∀𝑧((𝑅 FrSe 𝐴𝑋𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → (𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑧(𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → (𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))))
11 19.21v 1865 . . . . . . . . 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))))
1211imbi2i 326 . . . . . . . 8 (((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑧(𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → (𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))))
1310, 12bitri 264 . . . . . . 7 (∀𝑧((𝑅 FrSe 𝐴𝑋𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → (𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))))
1413albii 1744 . . . . . 6 (∀𝑦𝑧((𝑅 FrSe 𝐴𝑋𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → (𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))) ↔ ∀𝑦((𝑅 FrSe 𝐴𝑋𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))))
15 19.21v 1865 . . . . . 6 (∀𝑦((𝑅 FrSe 𝐴𝑋𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑦(𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))))
16 df-ral 2912 . . . . . . . 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))))
1716bicomi 214 . . . . . . 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)))
1817imbi2i 326 . . . . . 6 (((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑦(𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅)∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))))
1914, 15, 183bitri 286 . . . . 5 (∀𝑦𝑧((𝑅 FrSe 𝐴𝑋𝐴) → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → (𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅)∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))))
207, 9, 193bitri 286 . . . 4 (∀𝑦𝑧((𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅)∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))))
214, 20mpbi 220 . . 3 ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅)∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))
22 dfss2 3576 . . . 4 ( pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅) ↔ ∀𝑧(𝑧 ∈ pred(𝑦, 𝐴, 𝑅) → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)))
2322ralbii 2975 . . 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)))
2421, 23sylibr 224 . 2 ((𝑅 FrSe 𝐴𝑋𝐴) → ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
25 df-bnj19 30505 . 2 ( TrFo( trCl(𝑋, 𝐴, 𝑅), 𝐴, 𝑅) ↔ ∀𝑦 ∈ trCl (𝑋, 𝐴, 𝑅) pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
2624, 25sylibr 224 1 ((𝑅 FrSe 𝐴𝑋𝐴) → TrFo( trCl(𝑋, 𝐴, 𝑅), 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wal 1478  wcel 1987  wral 2907  wss 3559  w-bnj17 30494   predc-bnj14 30496   FrSe w-bnj15 30500   trClc-bnj18 30502   TrFow-bnj19 30504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-ral 2912  df-in 3566  df-ss 3573  df-bnj17 30495  df-bnj19 30505
This theorem is referenced by:  bnj907  30778
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