Theorem bnj965 35506

Description: Technical lemma for bnj852 35485. 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
bnj965.1	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj965.2	⊢ (𝜓″ ↔ [𝐺 / 𝑓]𝜓)
bnj965.12000	⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
bnj965.13000	⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})

Assertion

Ref	Expression
bnj965	⊢ (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Distinct variable groups:   𝐴,𝑓   𝑖,𝐺   𝑓,𝑁   𝑅,𝑓   𝑓,𝑖,𝑦   𝑦,𝑛

Allowed substitution hints:   𝜓(𝑦, 𝑓, 𝑖, 𝑚, 𝑛)   𝐴(𝑦, 𝑖, 𝑚, 𝑛)   𝐶(𝑦, 𝑓, 𝑖, 𝑚, 𝑛)   𝑅(𝑦, 𝑖, 𝑚, 𝑛)   𝐺(𝑦, 𝑓, 𝑚, 𝑛)   𝑁(𝑦, 𝑖, 𝑚, 𝑛)   𝜓″(𝑦, 𝑓, 𝑖, 𝑚, 𝑛)

Proof of Theorem bnj965

Step	Hyp	Ref	Expression
1		bnj965.1	. 2 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
2		bnj965.2	. 2 ⊢ (𝜓″ ↔ [𝐺 / 𝑓]𝜓)
3		bnj965.13000	. . 3 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
4	3	bnj918 35331	. 2 ⊢ 𝐺 ∈ V
5		bnj965.12000	. 2 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
6	1, 2, 4, 5, 3	bnj1000 35505	1 ⊢ (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145  ∀wral 3076  [wsbc 3738   ∪ cun 3896  {csn 4583  ⟨cop 4589  ∪ ciun 4950  suc csuc 6353  ‘cfv 6527  ωcom 7860   predc-bnj14 35253

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-iota 6483  df-fv 6535

This theorem is used by:  bnj964  35507  bnj999  35522