Theorem bnj965 34919

Description: Technical lemma for bnj852 34898. 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 34743	. 2 ⊢ 𝐺 ∈ V
5		bnj965.12000	. 2 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
6	1, 2, 4, 5, 3	bnj1000 34918	1 ⊢ (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2108  ∀wral 3051  [wsbc 3765   ∪ cun 3924  {csn 4601  ⟨cop 4607  ∪ ciun 4967  suc csuc 6354  ‘cfv 6530  ωcom 7859   predc-bnj14 34665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7727

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-iota 6483  df-fv 6538

This theorem is referenced by:  bnj964  34920  bnj999  34935