Theorem bnj981 35408

Description: Technical lemma for bnj69 35468. 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
bnj981.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj981.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj981.3	⊢ 𝐷 = (ω ∖ {∅})
bnj981.4	⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
bnj981.5	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))

Assertion

Ref	Expression
bnj981	⊢ (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖)))

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

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

Proof of Theorem bnj981

Step	Hyp	Ref	Expression
1		nfv 1947	. . . 4 ⊢ Ⅎ𝑦 𝑍 ∈ trCl(𝑋, 𝐴, 𝑅)
2		bnj981.2	. . . . . . . . . . . 12 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
3		nfcv 2927	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦ω
4		nfv 1947	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦 suc 𝑖 ∈ 𝑛
5		nfiu1 4994	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑦∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)
6	5	nfeq2 2944	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦(𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)
7	4, 6	nfim 1929	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦(suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
8	3, 7	nfralw 3314	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
9	2, 8	nfxfr 1886	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝜓
10	9	nf5ri 2234	. . . . . . . . . 10 ⊢ (𝜓 → ∀𝑦𝜓)
11		bnj981.5	. . . . . . . . . 10 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
12	10, 11	bnj1096 35241	. . . . . . . . 9 ⊢ (𝜒 → ∀𝑦𝜒)
13	12	nf5i 2184	. . . . . . . 8 ⊢ Ⅎ𝑦𝜒
14		nfv 1947	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑖 ∈ 𝑛
15		nfv 1947	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑍 ∈ (𝑓‘𝑖)
16	13, 14, 15	nf3an 1934	. . . . . . 7 ⊢ Ⅎ𝑦(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖))
17	16	nfex 2359	. . . . . 6 ⊢ Ⅎ𝑦∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖))
18	17	nfex 2359	. . . . 5 ⊢ Ⅎ𝑦∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖))
19	18	nfex 2359	. . . 4 ⊢ Ⅎ𝑦∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖))
20	1, 19	nfim 1929	. . 3 ⊢ Ⅎ𝑦(𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖)))
21		eleq1 2853	. . . 4 ⊢ (𝑦 = 𝑍 → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ 𝑍 ∈ trCl(𝑋, 𝐴, 𝑅)))
22		eleq1 2853	. . . . . 6 ⊢ (𝑦 = 𝑍 → (𝑦 ∈ (𝑓‘𝑖) ↔ 𝑍 ∈ (𝑓‘𝑖)))
23	22	3anbi3d 1470	. . . . 5 ⊢ (𝑦 = 𝑍 → ((𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)) ↔ (𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖))))
24	23	3exbidv 1958	. . . 4 ⊢ (𝑦 = 𝑍 → (∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)) ↔ ∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖))))
25	21, 24	imbi12d 347	. . 3 ⊢ (𝑦 = 𝑍 → ((𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖))) ↔ (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖)))))
26		bnj981.1	. . . 4 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
27		bnj981.3	. . . 4 ⊢ 𝐷 = (ω ∖ {∅})
28		bnj981.4	. . . 4 ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
29	26, 2, 27, 28, 11	bnj917 35392	. . 3 ⊢ (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)))
30	20, 25, 29	vtoclg1f 3537	. 2 ⊢ (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) → (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖))))
31	30	pm2.43i 53	1 ⊢ (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2146  {cab 2743  ∀wral 3081  ∃wrex 3091   ∖ cdif 3903  ∅c0 4286  {csn 4591  ∪ ciun 4958  suc csuc 6367   Fn wfn 6536  ‘cfv 6541  ωcom 7869   ∧ w-bnj17 35145   predc-bnj14 35147   trClc-bnj18 35153

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-v 3459  df-iun 4960  df-fn 6544  df-bnj17 35146  df-bnj18 35154

This theorem is used by:  bnj1128  35448