Theorem bnj911 32483

Description: Technical lemma for bnj69 32561. 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 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Assertion

Ref	Expression
bnj911	⊢ ((𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) → ∀𝑖(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))

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

Step	Hyp	Ref	Expression
1		bnj911.2	. . 3 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
2	1	bnj1095 32332	. 2 ⊢ (𝜓 → ∀𝑖𝜓)
3	2	bnj1350 32376	1 ⊢ ((𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) → ∀𝑖(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1088  ∀wal 1540   = wceq 1542   ∈ wcel 2114  ∀wral 3053  ∅c0 4211  ∪ ciun 4881  suc csuc 6174   Fn wfn 6334  ‘cfv 6339  ωcom 7599   predc-bnj14 32237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-10 2145  ax-12 2179

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-nf 1791  df-ral 3058

This theorem is referenced by:  bnj916  32484  bnj1014  32512