Theorem bnj981 34965

Description: Technical lemma for bnj69 35025. 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 1913	. . . 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 2904	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦ω
4		nfv 1913	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦 suc 𝑖 ∈ 𝑛
5		nfiu1 5026	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑦∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)
6	5	nfeq2 2922	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦(𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)
7	4, 6	nfim 1895	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦(suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
8	3, 7	nfralw 3310	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
9	2, 8	nfxfr 1852	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝜓
10	9	nf5ri 2194	. . . . . . . . . 10 ⊢ (𝜓 → ∀𝑦𝜓)
11		bnj981.5	. . . . . . . . . 10 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
12	10, 11	bnj1096 34797	. . . . . . . . 9 ⊢ (𝜒 → ∀𝑦𝜒)
13	12	nf5i 2145	. . . . . . . 8 ⊢ Ⅎ𝑦𝜒
14		nfv 1913	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑖 ∈ 𝑛
15		nfv 1913	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑍 ∈ (𝑓‘𝑖)
16	13, 14, 15	nf3an 1900	. . . . . . 7 ⊢ Ⅎ𝑦(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖))
17	16	nfex 2323	. . . . . 6 ⊢ Ⅎ𝑦∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖))
18	17	nfex 2323	. . . . 5 ⊢ Ⅎ𝑦∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖))
19	18	nfex 2323	. . . 4 ⊢ Ⅎ𝑦∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖))
20	1, 19	nfim 1895	. . 3 ⊢ Ⅎ𝑦(𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖)))
21		eleq1 2828	. . . 4 ⊢ (𝑦 = 𝑍 → (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ 𝑍 ∈ trCl(𝑋, 𝐴, 𝑅)))
22		eleq1 2828	. . . . . 6 ⊢ (𝑦 = 𝑍 → (𝑦 ∈ (𝑓‘𝑖) ↔ 𝑍 ∈ (𝑓‘𝑖)))
23	22	3anbi3d 1443	. . . . 5 ⊢ (𝑦 = 𝑍 → ((𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)) ↔ (𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖))))
24	23	3exbidv 1924	. . . 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 344	. . 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 34949	. . 3 ⊢ (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)))
30	20, 25, 29	vtoclg1f 3569	. 2 ⊢ (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) → (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖))))
31	30	pm2.43i 52	1 ⊢ (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086   = wceq 1539  ∃wex 1778   ∈ wcel 2107  {cab 2713  ∀wral 3060  ∃wrex 3069   ∖ cdif 3947  ∅c0 4332  {csn 4625  ∪ ciun 4990  suc csuc 6385   Fn wfn 6555  ‘cfv 6560  ωcom 7888   ∧ w-bnj17 34701   predc-bnj14 34703   trClc-bnj18 34709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-v 3481  df-iun 4992  df-fn 6563  df-bnj17 34702  df-bnj18 34710

This theorem is referenced by:  bnj1128  35005