Theorem bnj600 32301

Description: Technical lemma for bnj852 32303. 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 32271	. . . . . 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 3444	. . . . . . 7 ⊢ 𝑚 ∈ V
13	8, 9, 10, 11, 12	bnj207 32263	. . . . . 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 32299	. . . . . 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 32300	. . . . . 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 32110	. . . . . . . . 9 ⊢ ((𝑛 ≠ 1o ∧ 𝑛 ∈ 𝐷) → ∃𝑚 ∈ 𝐷 𝑛 = suc 𝑚)
20		df-rex 3112	. . . . . . . . 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 32109	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ 𝐷 → ∃𝑝 ∈ ω 𝑚 = suc 𝑝)
23		df-rex 3112	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚) → ∃𝑝(𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
26	25	ancri 553	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚) → (∃𝑝(𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ∧ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚)))
27	26	bnj534 32120	. . . . . . . . . 10 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚) → ∃𝑝((𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ∧ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚)))
28		bnj432 32096	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ ((𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ∧ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚)))
29	28	exbii 1849	. . . . . . . . . 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 1836	. . . . . . . 8 ⊢ (∃𝑚(𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚) → ∃𝑚∃𝑝(𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
32	21, 31	syl 17	. . . . . . 7 ⊢ ((𝑛 ≠ 1o ∧ 𝑛 ∈ 𝐷) → ∃𝑚∃𝑝(𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
33	5	2exbii 1850	. . . . . . 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 3170	. . . . . . . . 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 1108	. . . . . . 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 32269	. . . . . . . 8 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
41	16, 10, 12	bnj539 32273	. . . . . . . 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 32276	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝐺 Fn 𝑛)
43	39, 5, 42	bnj561 32285	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → 𝐺 Fn 𝑛)
44	40, 18, 6, 4, 39, 42, 15	bnj545 32277	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝜑″)
45	39, 5, 44	bnj562 32286	. . . . . 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 32288	. . . . . 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 32298	. . . . 5 ⊢ ((𝑛 ≠ 1o ∧ 𝑛 ∈ 𝐷 ∧ 𝜃) → ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
59	14, 16, 18	bnj579 32296	. . . . . . 7 ⊢ (𝑛 ∈ 𝐷 → ∃*𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
60	59	a1d 25	. . . . . 6 ⊢ (𝑛 ∈ 𝐷 → ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃*𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
61	60	3ad2ant2 1131	. . . . 5 ⊢ ((𝑛 ≠ 1o ∧ 𝑛 ∈ 𝐷 ∧ 𝜃) → ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → ∃*𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
62	58, 61	jcad 516	. . . 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 2629	. . . 4 ⊢ (∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (∃𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ ∃*𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
64	62, 63	syl6ibr 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 1119	1 ⊢ (𝑛 ≠ 1o → ((𝑛 ∈ 𝐷 ∧ 𝜃) → 𝜒))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃*wmo 2596  ∃!weu 2628   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  [wsbc 3720   ∖ cdif 3878   ∪ cun 3879  ∅c0 4243  {csn 4525  ⟨cop 4531  ∪ ciun 4881   class class class wbr 5030   E cep 5429  suc csuc 6161   Fn wfn 6319  ‘cfv 6324  ωcom 7560  1_oc1o 8078   ∧ w-bnj17 32066   predc-bnj14 32068   FrSe w-bnj15 32072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-reg 9040

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-om 7561  df-1o 8085  df-bnj17 32067  df-bnj14 32069  df-bnj13 32071  df-bnj15 32073

This theorem is referenced by:  bnj601  32302