Theorem bnj600 35415

Description: Technical lemma for bnj852 35417. 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
bnj600.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj600.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj600.3	⊢ 𝐷 = (ω ∖ {∅})
bnj600.4	⊢ (𝜒 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
bnj600.5	⊢ (𝜃 ↔ ∀𝑚 ∈ 𝐷 (𝑚 E 𝑛 → [𝑚 / 𝑛]𝜒))
bnj600.10	⊢ (𝜑′ ↔ [𝑚 / 𝑛]𝜑)
bnj600.11	⊢ (𝜓′ ↔ [𝑚 / 𝑛]𝜓)
bnj600.12	⊢ (𝜒′ ↔ [𝑚 / 𝑛]𝜒)
bnj600.13	⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑)
bnj600.14	⊢ (𝜓″ ↔ [𝐺 / 𝑓]𝜓)
bnj600.15	⊢ (𝜒″ ↔ [𝐺 / 𝑓]𝜒)
bnj600.16	⊢ 𝐺 = (𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
bnj600.17	⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
bnj600.18	⊢ (𝜎 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
bnj600.19	⊢ (𝜂 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
bnj600.20	⊢ (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 = suc 𝑖))
bnj600.21	⊢ (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 ≠ suc 𝑖))
bnj600.22	⊢ 𝐵 = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)
bnj600.23	⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)
bnj600.24	⊢ 𝐾 = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)
bnj600.25	⊢ 𝐿 = ∪ 𝑦 ∈ (𝐺‘𝑝) pred(𝑦, 𝐴, 𝑅)
bnj600.26	⊢ 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})

Assertion

Ref	Expression
bnj600	⊢ (𝑛 ≠ 1o → ((𝑛 ∈ 𝐷 ∧ 𝜃) → 𝜒))

Distinct variable groups:   𝐴,𝑓,𝑖,𝑚,𝑛,𝑝   𝑦,𝐴,𝑓,𝑖,𝑛,𝑝   𝐷,𝑓,𝑝   𝑖,𝐺,𝑦   𝑅,𝑓,𝑖,𝑚,𝑛,𝑝   𝑦,𝑅   𝜂,𝑓,𝑖   𝑥,𝑓,𝑚,𝑛,𝑝   𝑖,𝜑′,𝑝   𝜑,𝑚,𝑝   𝜓,𝑚,𝑝   𝜃,𝑝

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑓, 𝑖, 𝑛)   𝜓(𝑥, 𝑦, 𝑓, 𝑖, 𝑛)   𝜒(𝑥, 𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝜃(𝑥, 𝑦, 𝑓, 𝑖, 𝑚, 𝑛)   𝜏(𝑥, 𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝜂(𝑥, 𝑦, 𝑚, 𝑛, 𝑝)   𝜁(𝑥, 𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝜎(𝑥, 𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝜌(𝑥, 𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝐴(𝑥)   𝐵(𝑥, 𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝐶(𝑥, 𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝐷(𝑥, 𝑦, 𝑖, 𝑚, 𝑛)   𝑅(𝑥)   𝐺(𝑥, 𝑓, 𝑚, 𝑛, 𝑝)   𝐾(𝑥, 𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝐿(𝑥, 𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝜑′(𝑥, 𝑦, 𝑓, 𝑚, 𝑛)   𝜓′(𝑥, 𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝜒′(𝑥, 𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝜑″(𝑥, 𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝜓″(𝑥, 𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝜒″(𝑥, 𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)

Proof of Theorem bnj600

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj600.5	. . . . . 6 ⊢ (𝜃 ↔ ∀𝑚 ∈ 𝐷 (𝑚 E 𝑛 → [𝑚 / 𝑛]𝜒))
2		bnj600.13	. . . . . 6 ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑)
3		bnj600.14	. . . . . 6 ⊢ (𝜓″ ↔ [𝐺 / 𝑓]𝜓)
4		bnj600.17	. . . . . 6 ⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
5		bnj600.19	. . . . . 6 ⊢ (𝜂 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
6		bnj600.16	. . . . . . 7 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
7	6	bnj528 35385	. . . . . 6 ⊢ 𝐺 ∈ V
8		bnj600.4	. . . . . . 7 ⊢ (𝜒 ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
9		bnj600.10	. . . . . . 7 ⊢ (𝜑′ ↔ [𝑚 / 𝑛]𝜑)
10		bnj600.11	. . . . . . 7 ⊢ (𝜓′ ↔ [𝑚 / 𝑛]𝜓)
11		bnj600.12	. . . . . . 7 ⊢ (𝜒′ ↔ [𝑚 / 𝑛]𝜒)
12		vex 3457	. . . . . . 7 ⊢ 𝑚 ∈ V
13	8, 9, 10, 11, 12	bnj207 35377	. . . . . 6 ⊢ (𝜒′ ↔ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′)))
14		bnj600.1	. . . . . . 7 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
15	14, 2, 7	bnj609 35413	. . . . . 6 ⊢ (𝜑″ ↔ (𝐺‘∅) = pred(𝑥, 𝐴, 𝑅))
16		bnj600.2	. . . . . . 7 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
17	16, 3, 7	bnj611 35414	. . . . . 6 ⊢ (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)))
18		bnj600.3	. . . . . . . . . 10 ⊢ 𝐷 = (ω ∖ {∅})
19	18	bnj168 35227	. . . . . . . . 9 ⊢ ((𝑛 ≠ 1o ∧ 𝑛 ∈ 𝐷) → ∃𝑚 ∈ 𝐷 𝑛 = suc 𝑚)
20		df-rex 3089	. . . . . . . . 9 ⊢ (∃𝑚 ∈ 𝐷 𝑛 = suc 𝑚 ↔ ∃𝑚(𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚))
21	19, 20	sylib 221	. . . . . . . 8 ⊢ ((𝑛 ≠ 1o ∧ 𝑛 ∈ 𝐷) → ∃𝑚(𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚))
22	18	bnj158 35226	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ 𝐷 → ∃𝑝 ∈ ω 𝑚 = suc 𝑝)
23		df-rex 3089	. . . . . . . . . . . . . 14 ⊢ (∃𝑝 ∈ ω 𝑚 = suc 𝑝 ↔ ∃𝑝(𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
24	22, 23	sylib 221	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ 𝐷 → ∃𝑝(𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
25	24	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚) → ∃𝑝(𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
26	25	ancri 559	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚) → (∃𝑝(𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ∧ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚)))
27	26	bnj534 35236	. . . . . . . . . 10 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚) → ∃𝑝((𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ∧ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚)))
28		bnj432 35213	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ ((𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ∧ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚)))
29	28	exbii 1881	. . . . . . . . . 10 ⊢ (∃𝑝(𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ ∃𝑝((𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ∧ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚)))
30	27, 29	sylibr 237	. . . . . . . . 9 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚) → ∃𝑝(𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
31	30	eximi 1868	. . . . . . . 8 ⊢ (∃𝑚(𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚) → ∃𝑚∃𝑝(𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
32	21, 31	syl 18	. . . . . . 7 ⊢ ((𝑛 ≠ 1o ∧ 𝑛 ∈ 𝐷) → ∃𝑚∃𝑝(𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
33	5	2exbii 1882	. . . . . . 7 ⊢ (∃𝑚∃𝑝𝜂 ↔ ∃𝑚∃𝑝(𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
34	32, 33	sylibr 237	. . . . . 6 ⊢ ((𝑛 ≠ 1o ∧ 𝑛 ∈ 𝐷) → ∃𝑚∃𝑝𝜂)
35		rsp 3252	. . . . . . . . 9 ⊢ (∀𝑚 ∈ 𝐷 (𝑚 E 𝑛 → [𝑚 / 𝑛]𝜒) → (𝑚 ∈ 𝐷 → (𝑚 E 𝑛 → [𝑚 / 𝑛]𝜒)))
36	1, 35	sylbi 220	. . . . . . . 8 ⊢ (𝜃 → (𝑚 ∈ 𝐷 → (𝑚 E 𝑛 → [𝑚 / 𝑛]𝜒)))
37	36	3imp 1128	. . . . . . 7 ⊢ ((𝜃 ∧ 𝑚 ∈ 𝐷 ∧ 𝑚 E 𝑛) → [𝑚 / 𝑛]𝜒)
38	37, 11	sylibr 237	. . . . . 6 ⊢ ((𝜃 ∧ 𝑚 ∈ 𝐷 ∧ 𝑚 E 𝑛) → 𝜒′)
39		bnj600.18	. . . . . . 7 ⊢ (𝜎 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
40	14, 9, 12	bnj523 35383	. . . . . . . 8 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
41	16, 10, 12	bnj539 35387	. . . . . . . 8 ⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
42	40, 41, 18, 6, 4, 39	bnj544 35390	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝐺 Fn 𝑛)
43	39, 5, 42	bnj561 35399	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → 𝐺 Fn 𝑛)
44	40, 18, 6, 4, 39, 42, 15	bnj545 35391	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝜑″)
45	39, 5, 44	bnj562 35400	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → 𝜑″)
46		bnj600.20	. . . . . . 7 ⊢ (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 = suc 𝑖))
47		bnj600.22	. . . . . . 7 ⊢ 𝐵 = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)
48		bnj600.23	. . . . . . 7 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)
49		bnj600.24	. . . . . . 7 ⊢ 𝐾 = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)
50		bnj600.25	. . . . . . 7 ⊢ 𝐿 = ∪ 𝑦 ∈ (𝐺‘𝑝) pred(𝑦, 𝐴, 𝑅)
51		bnj600.26	. . . . . . 7 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
52		bnj600.21	. . . . . . 7 ⊢ (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 ≠ suc 𝑖))
53	18, 6, 4, 39, 5, 46, 47, 48, 49, 50, 51, 40, 41, 42, 52, 43, 17	bnj571 35402	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → 𝜓″)
54		biid 264	. . . . . 6 ⊢ ([𝑧 / 𝑓]𝜑 ↔ [𝑧 / 𝑓]𝜑)
55		biid 264	. . . . . 6 ⊢ ([𝑧 / 𝑓]𝜓 ↔ [𝑧 / 𝑓]𝜓)
56		biid 264	. . . . . 6 ⊢ ([𝐺 / 𝑧][𝑧 / 𝑓]𝜑 ↔ [𝐺 / 𝑧][𝑧 / 𝑓]𝜑)
57		biid 264	. . . . . 6 ⊢ ([𝐺 / 𝑧][𝑧 / 𝑓]𝜓 ↔ [𝐺 / 𝑧][𝑧 / 𝑓]𝜓)
58	1, 2, 3, 4, 5, 7, 13, 15, 17, 34, 38, 43, 45, 53, 14, 16, 54, 55, 56, 57	bnj607 35412	. . . . 5 ⊢ ((𝑛 ≠ 1o ∧ 𝑛 ∈ 𝐷 ∧ 𝜃) → ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
59	14, 16, 18	bnj579 35410	. . . . . . 7 ⊢ (𝑛 ∈ 𝐷 → ∃*𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
60	59	a1d 26	. . . . . 6 ⊢ (𝑛 ∈ 𝐷 → ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃*𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
61	60	3ad2ant2 1152	. . . . 5 ⊢ ((𝑛 ≠ 1o ∧ 𝑛 ∈ 𝐷 ∧ 𝜃) → ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃*𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
62	58, 61	jcad 522	. . . 4 ⊢ ((𝑛 ≠ 1o ∧ 𝑛 ∈ 𝐷 ∧ 𝜃) → ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → (∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ ∃*𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))))
63		df-eu 2596	. . . 4 ⊢ (∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ ∃*𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
64	62, 63	imbitrrdi 255	. . 3 ⊢ ((𝑛 ≠ 1o ∧ 𝑛 ∈ 𝐷 ∧ 𝜃) → ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
65	64, 8	sylibr 237	. 2 ⊢ ((𝑛 ≠ 1o ∧ 𝑛 ∈ 𝐷 ∧ 𝜃) → 𝜒)
66	65	3expib 1140	1 ⊢ (𝑛 ≠ 1o → ((𝑛 ∈ 𝐷 ∧ 𝜃) → 𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃*wmo 2564  ∃!weu 2595   ≠ wne 2957  ∀wral 3078  ∃wrex 3088  [wsbc 3742   ∖ cdif 3899   ∪ cun 3900  ∅c0 4282  {csn 4587  ⟨cop 4593  ∪ ciun 4954   class class class wbr 5107   E cep 5558  suc csuc 6363   Fn wfn 6532  ‘cfv 6537  ωcom 7865  1_oc1o 8451   ∧ w-bnj17 35183   predc-bnj14 35185   FrSe w-bnj15 35189

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7739  ax-reg 9567

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-fv 6545  df-om 7866  df-1o 8458  df-bnj17 35184  df-bnj14 35186  df-bnj13 35188  df-bnj15 35190

This theorem is used by:  bnj601  35416