Theorem bnj983 32931

Description: Technical lemma for bnj69 32990. 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
bnj983.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj983.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj983.3	⊢ 𝐷 = (ω ∖ {∅})
bnj983.4	⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
bnj983.5	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))

Assertion

Ref	Expression
bnj983	⊢ (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ ∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖)))

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

Proof of Theorem bnj983

Step	Hyp	Ref	Expression
1		bnj983.1	. . . . . . . 8 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2		bnj983.2	. . . . . . . 8 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
3		bnj983.3	. . . . . . . 8 ⊢ 𝐷 = (ω ∖ {∅})
4		bnj983.4	. . . . . . . 8 ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
5	1, 2, 3, 4	bnj882 32906	. . . . . . 7 ⊢ trCl(𝑋, 𝐴, 𝑅) = ∪ 𝑓 ∈ 𝐵 ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖)
6	5	eleq2i 2830	. . . . . 6 ⊢ (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ 𝑍 ∈ ∪ 𝑓 ∈ 𝐵 ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖))
7		eliun 4928	. . . . . . 7 ⊢ (𝑍 ∈ ∪ 𝑓 ∈ 𝐵 ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ↔ ∃𝑓 ∈ 𝐵 𝑍 ∈ ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖))
8		eliun 4928	. . . . . . . 8 ⊢ (𝑍 ∈ ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ↔ ∃𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓‘𝑖))
9	8	rexbii 3181	. . . . . . 7 ⊢ (∃𝑓 ∈ 𝐵 𝑍 ∈ ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ↔ ∃𝑓 ∈ 𝐵 ∃𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓‘𝑖))
10	7, 9	bitri 274	. . . . . 6 ⊢ (𝑍 ∈ ∪ 𝑓 ∈ 𝐵 ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ↔ ∃𝑓 ∈ 𝐵 ∃𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓‘𝑖))
11		df-rex 3070	. . . . . . 7 ⊢ (∃𝑓 ∈ 𝐵 ∃𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓‘𝑖) ↔ ∃𝑓(𝑓 ∈ 𝐵 ∧ ∃𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓‘𝑖)))
12	4	abeq2i 2875	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝐵 ↔ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
13	12	anbi1i 624	. . . . . . . 8 ⊢ ((𝑓 ∈ 𝐵 ∧ ∃𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓‘𝑖)) ↔ (∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ ∃𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓‘𝑖)))
14	13	exbii 1850	. . . . . . 7 ⊢ (∃𝑓(𝑓 ∈ 𝐵 ∧ ∃𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓‘𝑖)) ↔ ∃𝑓(∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ ∃𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓‘𝑖)))
15	11, 14	bitri 274	. . . . . 6 ⊢ (∃𝑓 ∈ 𝐵 ∃𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓‘𝑖) ↔ ∃𝑓(∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ ∃𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓‘𝑖)))
16	6, 10, 15	3bitri 297	. . . . 5 ⊢ (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ ∃𝑓(∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ ∃𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓‘𝑖)))
17		bnj983.5	. . . . . . . . 9 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
18		bnj252 32682	. . . . . . . . 9 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
19	17, 18	bitri 274	. . . . . . . 8 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
20	19	exbii 1850	. . . . . . 7 ⊢ (∃𝑛𝜒 ↔ ∃𝑛(𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
21	20	anbi1i 624	. . . . . 6 ⊢ ((∃𝑛𝜒 ∧ ∃𝑖(𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖))) ↔ (∃𝑛(𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ∧ ∃𝑖(𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖))))
22		df-rex 3070	. . . . . . 7 ⊢ (∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ∃𝑛(𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
23		df-rex 3070	. . . . . . 7 ⊢ (∃𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓‘𝑖) ↔ ∃𝑖(𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖)))
24	22, 23	anbi12i 627	. . . . . 6 ⊢ ((∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ ∃𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓‘𝑖)) ↔ (∃𝑛(𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ∧ ∃𝑖(𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖))))
25	21, 24	bitr4i 277	. . . . 5 ⊢ ((∃𝑛𝜒 ∧ ∃𝑖(𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖))) ↔ (∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ ∃𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓‘𝑖)))
26	16, 25	bnj133 32706	. . . 4 ⊢ (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ ∃𝑓(∃𝑛𝜒 ∧ ∃𝑖(𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖))))
27		19.41v 1953	. . . 4 ⊢ (∃𝑛(𝜒 ∧ ∃𝑖(𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖))) ↔ (∃𝑛𝜒 ∧ ∃𝑖(𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖))))
28	26, 27	bnj133 32706	. . 3 ⊢ (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ ∃𝑓∃𝑛(𝜒 ∧ ∃𝑖(𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖))))
29	2	bnj1095 32761	. . . . . . 7 ⊢ (𝜓 → ∀𝑖𝜓)
30	29, 17	bnj1096 32762	. . . . . 6 ⊢ (𝜒 → ∀𝑖𝜒)
31	30	nf5i 2142	. . . . 5 ⊢ Ⅎ𝑖𝜒
32	31	19.42 2229	. . . 4 ⊢ (∃𝑖(𝜒 ∧ (𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖))) ↔ (𝜒 ∧ ∃𝑖(𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖))))
33	32	2exbii 1851	. . 3 ⊢ (∃𝑓∃𝑛∃𝑖(𝜒 ∧ (𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖))) ↔ ∃𝑓∃𝑛(𝜒 ∧ ∃𝑖(𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖))))
34	28, 33	bitr4i 277	. 2 ⊢ (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ ∃𝑓∃𝑛∃𝑖(𝜒 ∧ (𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖))))
35		3anass 1094	. . 3 ⊢ ((𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖)) ↔ (𝜒 ∧ (𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖))))
36	35	3exbii 1852	. 2 ⊢ (∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖)) ↔ ∃𝑓∃𝑛∃𝑖(𝜒 ∧ (𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖))))
37		fndm 6536	. . . . . . . 8 ⊢ (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
38	17, 37	bnj770 32743	. . . . . . 7 ⊢ (𝜒 → dom 𝑓 = 𝑛)
39		eleq2 2827	. . . . . . . 8 ⊢ (dom 𝑓 = 𝑛 → (𝑖 ∈ dom 𝑓 ↔ 𝑖 ∈ 𝑛))
40	39	3anbi2d 1440	. . . . . . 7 ⊢ (dom 𝑓 = 𝑛 → ((𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖)) ↔ (𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖))))
41	38, 40	syl 17	. . . . . 6 ⊢ (𝜒 → ((𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖)) ↔ (𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖))))
42	41	3ad2ant1 1132	. . . . 5 ⊢ ((𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖)) → ((𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖)) ↔ (𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖))))
43	42	ibi 266	. . . 4 ⊢ ((𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖)) → (𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖)))
44	41	3ad2ant1 1132	. . . . 5 ⊢ ((𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖)) → ((𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖)) ↔ (𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖))))
45	44	ibir 267	. . . 4 ⊢ ((𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖)) → (𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖)))
46	43, 45	impbii 208	. . 3 ⊢ ((𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖)) ↔ (𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖)))
47	46	3exbii 1852	. 2 ⊢ (∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖)) ↔ ∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖)))
48	34, 36, 47	3bitr2i 299	1 ⊢ (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ ∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539  ∃wex 1782   ∈ wcel 2106  {cab 2715  ∀wral 3064  ∃wrex 3065   ∖ cdif 3884  ∅c0 4256  {csn 4561  ∪ ciun 4924  dom cdm 5589  suc csuc 6268   Fn wfn 6428  ‘cfv 6433  ωcom 7712   ∧ w-bnj17 32665   predc-bnj14 32667   trClc-bnj18 32673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-v 3434  df-iun 4926  df-fn 6436  df-bnj17 32666  df-bnj18 32674

This theorem is referenced by:  bnj1033  32949