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Theorem bnj561 34534
Description: Technical lemma for bnj852 34552. 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
bnj561.18 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
bnj561.19 (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
bnj561.37 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
Assertion
Ref Expression
bnj561 ((𝑅 FrSe 𝐴𝜏𝜂) → 𝐺 Fn 𝑛)

Proof of Theorem bnj561
StepHypRef Expression
1 bnj561.18 . . 3 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
2 bnj561.19 . . 3 (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
31, 2bnj556 34531 . 2 (𝜂𝜎)
4 bnj561.37 . 2 ((𝑅 FrSe 𝐴𝜏𝜎) → 𝐺 Fn 𝑛)
53, 4syl3an3 1163 1 ((𝑅 FrSe 𝐴𝜏𝜂) → 𝐺 Fn 𝑛)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1085   = wceq 1534  wcel 2099  suc csuc 6371   Fn wfn 6543  ωcom 7870  w-bnj17 34317   FrSe w-bnj15 34323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3473  df-un 3952  df-sn 4630  df-suc 6375  df-bnj17 34318
This theorem is referenced by:  bnj600  34550  bnj908  34562
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