Theorem bnj983 35648

Description: Technical lemma for bnj69 35707. 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 35623	. . . . . . 7 ⊢ trCl(𝑋, 𝐴, 𝑅) = ∪ 𝑓 ∈ 𝐵 ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖)
6	5	eleq2i 2856	. . . . . 6 ⊢ (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ 𝑍 ∈ ∪ 𝑓 ∈ 𝐵 ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖))
7		eliun 4958	. . . . . . 7 ⊢ (𝑍 ∈ ∪ 𝑓 ∈ 𝐵 ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ↔ ∃𝑓 ∈ 𝐵 𝑍 ∈ ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖))
8		eliun 4958	. . . . . . . 8 ⊢ (𝑍 ∈ ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ↔ ∃𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓‘𝑖))
9	8	rexbii 3113	. . . . . . 7 ⊢ (∃𝑓 ∈ 𝐵 𝑍 ∈ ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ↔ ∃𝑓 ∈ 𝐵 ∃𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓‘𝑖))
10	7, 9	bitri 278	. . . . . 6 ⊢ (𝑍 ∈ ∪ 𝑓 ∈ 𝐵 ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ↔ ∃𝑓 ∈ 𝐵 ∃𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓‘𝑖))
11		df-rex 3091	. . . . . . 7 ⊢ (∃𝑓 ∈ 𝐵 ∃𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓‘𝑖) ↔ ∃𝑓(𝑓 ∈ 𝐵 ∧ ∃𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓‘𝑖)))
12	4	eqabri 2906	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝐵 ↔ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
13	12	anbi1i 636	. . . . . . . 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 1881	. . . . . . 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 278	. . . . . 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 300	. . . . 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 35401	. . . . . . . . 9 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
19	17, 18	bitri 278	. . . . . . . 8 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
20	19	exbii 1881	. . . . . . 7 ⊢ (∃𝑛𝜒 ↔ ∃𝑛(𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
21	20	anbi1i 636	. . . . . 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 3091	. . . . . . 7 ⊢ (∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ∃𝑛(𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
23		df-rex 3091	. . . . . . 7 ⊢ (∃𝑖 ∈ dom 𝑓 𝑍 ∈ (𝑓‘𝑖) ↔ ∃𝑖(𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖)))
24	22, 23	anbi12i 640	. . . . . 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 281	. . . . 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 35425	. . . 4 ⊢ (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ ∃𝑓(∃𝑛𝜒 ∧ ∃𝑖(𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖))))
27		19.41v 1984	. . . 4 ⊢ (∃𝑛(𝜒 ∧ ∃𝑖(𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖))) ↔ (∃𝑛𝜒 ∧ ∃𝑖(𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖))))
28	26, 27	bnj133 35425	. . 3 ⊢ (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ ∃𝑓∃𝑛(𝜒 ∧ ∃𝑖(𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖))))
29	2	bnj1095 35479	. . . . . . 7 ⊢ (𝜓 → ∀𝑖𝜓)
30	29, 17	bnj1096 35480	. . . . . 6 ⊢ (𝜒 → ∀𝑖𝜒)
31	30	nf5i 2186	. . . . 5 ⊢ Ⅎ𝑖𝜒
32	31	19.42 2276	. . . 4 ⊢ (∃𝑖(𝜒 ∧ (𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖))) ↔ (𝜒 ∧ ∃𝑖(𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖))))
33	32	2exbii 1882	. . 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 281	. 2 ⊢ (𝑍 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ ∃𝑓∃𝑛∃𝑖(𝜒 ∧ (𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖))))
35		3anass 1111	. . 3 ⊢ ((𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖)) ↔ (𝜒 ∧ (𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖))))
36	35	3exbii 1883	. 2 ⊢ (∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖)) ↔ ∃𝑓∃𝑛∃𝑖(𝜒 ∧ (𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖))))
37		fndm 6661	. . . . . . . 8 ⊢ (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
38	17, 37	bnj770 35461	. . . . . . 7 ⊢ (𝜒 → dom 𝑓 = 𝑛)
39		eleq2 2853	. . . . . . . 8 ⊢ (dom 𝑓 = 𝑛 → (𝑖 ∈ dom 𝑓 ↔ 𝑖 ∈ 𝑛))
40	39	3anbi2d 1470	. . . . . . 7 ⊢ (dom 𝑓 = 𝑛 → ((𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖)) ↔ (𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖))))
41	38, 40	syl 18	. . . . . 6 ⊢ (𝜒 → ((𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖)) ↔ (𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖))))
42	41	3ad2ant1 1151	. . . . 5 ⊢ ((𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖)) → ((𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖)) ↔ (𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖))))
43	42	ibi 270	. . . 4 ⊢ ((𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖)) → (𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖)))
44	41	3ad2ant1 1151	. . . . 5 ⊢ ((𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖)) → ((𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖)) ↔ (𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖))))
45	44	ibir 271	. . . 4 ⊢ ((𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖)) → (𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖)))
46	43, 45	impbii 212	. . 3 ⊢ ((𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑍 ∈ (𝑓‘𝑖)) ↔ (𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑍 ∈ (𝑓‘𝑖)))
47	46	3exbii 1883	. 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 302	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   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090   ∖ cdif 4048  ∅c0 4281  {csn 4588  ∪ ciun 4954  dom cdm 5655  suc csuc 6379   Fn wfn 6550  ‘cfv 6555  ωcom 7898   ∧ w-bnj17 35384   predc-bnj14 35386   trClc-bnj18 35392

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-12 2216  ax-ext 2735

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-v 3456  df-iun 4956  df-fn 6558  df-bnj17 35385  df-bnj18 35393

This theorem is used by:  bnj1033  35666