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Theorem bnj911 32912
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
bnj911.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj911.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
Assertion
Ref Expression
bnj911 ((𝑓 Fn 𝑛𝜑𝜓) → ∀𝑖(𝑓 Fn 𝑛𝜑𝜓))
Distinct variable groups:   𝑓,𝑖   𝑖,𝑛   𝜑,𝑖
Allowed substitution hints:   𝜑(𝑦,𝑓,𝑛)   𝜓(𝑦,𝑓,𝑖,𝑛)   𝐴(𝑦,𝑓,𝑖,𝑛)   𝑅(𝑦,𝑓,𝑖,𝑛)   𝑋(𝑦,𝑓,𝑖,𝑛)

Proof of Theorem bnj911
StepHypRef Expression
1 bnj911.2 . . 3 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
21bnj1095 32761 . 2 (𝜓 → ∀𝑖𝜓)
32bnj1350 32805 1 ((𝑓 Fn 𝑛𝜑𝜓) → ∀𝑖(𝑓 Fn 𝑛𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1086  wal 1537   = wceq 1539  wcel 2106  wral 3064  c0 4256   ciun 4924  suc csuc 6268   Fn wfn 6428  cfv 6433  ωcom 7712   predc-bnj14 32667
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-10 2137  ax-12 2171
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-ex 1783  df-nf 1787  df-ral 3069
This theorem is referenced by:  bnj916  32913  bnj1014  32941
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