Theorem bnj917 35564

Description: Technical lemma for bnj69 35640. 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
bnj917.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj917.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj917.3	⊢ 𝐷 = (ω ∖ {∅})
bnj917.4	⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
bnj917.5	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))

Assertion

Ref	Expression
bnj917	⊢ (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)))

Distinct variable groups:   𝐴,𝑓,𝑖,𝑛,𝑦   𝐷,𝑖   𝑅,𝑓,𝑖,𝑛,𝑦   𝑓,𝑋,𝑖,𝑛,𝑦   𝜑,𝑖

Allowed substitution hints:   𝜑(𝑦, 𝑓, 𝑛)   𝜓(𝑦, 𝑓, 𝑖, 𝑛)   𝜒(𝑦, 𝑓, 𝑖, 𝑛)   𝐵(𝑦, 𝑓, 𝑖, 𝑛)   𝐷(𝑦, 𝑓, 𝑛)

Proof of Theorem bnj917

Step	Hyp	Ref	Expression
1		bnj917.1	. . 3 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2		bnj917.2	. . 3 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
3		bnj917.3	. . 3 ⊢ 𝐷 = (ω ∖ {∅})
4		bnj917.4	. . 3 ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
5		biid 264	. . 3 ⊢ ((𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
6	1, 2, 3, 4, 5	bnj916 35563	. 2 ⊢ (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓∃𝑛∃𝑖(𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)))
7		bnj917.5	. . . . . 6 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
8		bnj252 35334	. . . . . 6 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
9	7, 8	bitri 278	. . . . 5 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
10	9	3anbi1i 1175	. . . 4 ⊢ ((𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)) ↔ ((𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)))
11		bnj253 35335	. . . 4 ⊢ ((𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)) ↔ ((𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)))
12	10, 11	bitr4i 281	. . 3 ⊢ ((𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)) ↔ (𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)))
13	12	3exbii 1882	. 2 ⊢ (∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)) ↔ ∃𝑓∃𝑛∃𝑖(𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)))
14	6, 13	sylibr 237	1 ⊢ (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569  ∃wex 1811   ∈ wcel 2144  {cab 2738  ∀wral 3076  ∃wrex 3086   ∖ cdif 4045  ∅c0 4278  {csn 4583  ∪ ciun 4950  suc csuc 6364   Fn wfn 6533  ‘cfv 6538  ωcom 7877   ∧ w-bnj17 35317   predc-bnj14 35319   trClc-bnj18 35325

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-v 3452  df-iun 4952  df-fn 6541  df-bnj17 35318  df-bnj18 35326

This theorem is used by:  bnj981  35580  bnj996  35586