Theorem bnj916 32913

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
bnj916.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj916.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj916.3	⊢ 𝐷 = (ω ∖ {∅})
bnj916.4	⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
bnj916.5	⊢ (𝜒 ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))

Assertion

Ref	Expression
bnj916	⊢ (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓∃𝑛∃𝑖(𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)))

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

Proof of Theorem bnj916

Step	Hyp	Ref	Expression
1		bnj256 32685	. . . . . 6 ⊢ ((𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑦 ∈ (𝑓‘𝑖)) ↔ ((𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ∧ (𝑖 ∈ dom 𝑓 ∧ 𝑦 ∈ (𝑓‘𝑖))))
2	1	2exbii 1851	. . . . 5 ⊢ (∃𝑛∃𝑖(𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑦 ∈ (𝑓‘𝑖)) ↔ ∃𝑛∃𝑖((𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ∧ (𝑖 ∈ dom 𝑓 ∧ 𝑦 ∈ (𝑓‘𝑖))))
3		19.41v 1953	. . . . . 6 ⊢ (∃𝑛((𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ∧ ∃𝑖(𝑖 ∈ dom 𝑓 ∧ 𝑦 ∈ (𝑓‘𝑖))) ↔ (∃𝑛(𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ∧ ∃𝑖(𝑖 ∈ dom 𝑓 ∧ 𝑦 ∈ (𝑓‘𝑖))))
4		nfv 1917	. . . . . . . . 9 ⊢ Ⅎ𝑖 𝑛 ∈ 𝐷
5		bnj916.1	. . . . . . . . . . 11 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
6		bnj916.2	. . . . . . . . . . 11 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
7	5, 6	bnj911 32912	. . . . . . . . . 10 ⊢ ((𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) → ∀𝑖(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
8	7	nf5i 2142	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)
9	4, 8	nfan 1902	. . . . . . . 8 ⊢ Ⅎ𝑖(𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
10	9	19.42 2229	. . . . . . 7 ⊢ (∃𝑖((𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ∧ (𝑖 ∈ dom 𝑓 ∧ 𝑦 ∈ (𝑓‘𝑖))) ↔ ((𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ∧ ∃𝑖(𝑖 ∈ dom 𝑓 ∧ 𝑦 ∈ (𝑓‘𝑖))))
11	10	exbii 1850	. . . . . 6 ⊢ (∃𝑛∃𝑖((𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ∧ (𝑖 ∈ dom 𝑓 ∧ 𝑦 ∈ (𝑓‘𝑖))) ↔ ∃𝑛((𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ∧ ∃𝑖(𝑖 ∈ dom 𝑓 ∧ 𝑦 ∈ (𝑓‘𝑖))))
12		df-rex 3070	. . . . . . 7 ⊢ (∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ∃𝑛(𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
13		df-rex 3070	. . . . . . 7 ⊢ (∃𝑖 ∈ dom 𝑓 𝑦 ∈ (𝑓‘𝑖) ↔ ∃𝑖(𝑖 ∈ dom 𝑓 ∧ 𝑦 ∈ (𝑓‘𝑖)))
14	12, 13	anbi12i 627	. . . . . 6 ⊢ ((∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ ∃𝑖 ∈ dom 𝑓 𝑦 ∈ (𝑓‘𝑖)) ↔ (∃𝑛(𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ∧ ∃𝑖(𝑖 ∈ dom 𝑓 ∧ 𝑦 ∈ (𝑓‘𝑖))))
15	3, 11, 14	3bitr4i 303	. . . . 5 ⊢ (∃𝑛∃𝑖((𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ∧ (𝑖 ∈ dom 𝑓 ∧ 𝑦 ∈ (𝑓‘𝑖))) ↔ (∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ ∃𝑖 ∈ dom 𝑓 𝑦 ∈ (𝑓‘𝑖)))
16	2, 15	bitri 274	. . . 4 ⊢ (∃𝑛∃𝑖(𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑦 ∈ (𝑓‘𝑖)) ↔ (∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ ∃𝑖 ∈ dom 𝑓 𝑦 ∈ (𝑓‘𝑖)))
17	16	exbii 1850	. . 3 ⊢ (∃𝑓∃𝑛∃𝑖(𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑦 ∈ (𝑓‘𝑖)) ↔ ∃𝑓(∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ ∃𝑖 ∈ dom 𝑓 𝑦 ∈ (𝑓‘𝑖)))
18		bnj916.5	. . . . . . 7 ⊢ (𝜒 ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
19	18	3anbi2i 1157	. . . . . 6 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑖 ∈ dom 𝑓) ↔ (𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ 𝑖 ∈ dom 𝑓))
20	19	anbi1i 624	. . . . 5 ⊢ (((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑖 ∈ dom 𝑓) ∧ 𝑦 ∈ (𝑓‘𝑖)) ↔ ((𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ 𝑖 ∈ dom 𝑓) ∧ 𝑦 ∈ (𝑓‘𝑖)))
21		df-bnj17 32666	. . . . 5 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑦 ∈ (𝑓‘𝑖)) ↔ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑖 ∈ dom 𝑓) ∧ 𝑦 ∈ (𝑓‘𝑖)))
22		df-bnj17 32666	. . . . 5 ⊢ ((𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑦 ∈ (𝑓‘𝑖)) ↔ ((𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ 𝑖 ∈ dom 𝑓) ∧ 𝑦 ∈ (𝑓‘𝑖)))
23	20, 21, 22	3bitr4i 303	. . . 4 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑦 ∈ (𝑓‘𝑖)) ↔ (𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑦 ∈ (𝑓‘𝑖)))
24	23	3exbii 1852	. . 3 ⊢ (∃𝑓∃𝑛∃𝑖(𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑦 ∈ (𝑓‘𝑖)) ↔ ∃𝑓∃𝑛∃𝑖(𝑛 ∈ 𝐷 ∧ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑦 ∈ (𝑓‘𝑖)))
25		bnj916.3	. . . . . . 7 ⊢ 𝐷 = (ω ∖ {∅})
26		bnj916.4	. . . . . . 7 ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
27	5, 6, 25, 26	bnj882 32906	. . . . . 6 ⊢ trCl(𝑋, 𝐴, 𝑅) = ∪ 𝑓 ∈ 𝐵 ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖)
28	27	eleq2i 2830	. . . . 5 ⊢ (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ 𝑦 ∈ ∪ 𝑓 ∈ 𝐵 ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖))
29		eliun 4928	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑓 ∈ 𝐵 ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ↔ ∃𝑓 ∈ 𝐵 𝑦 ∈ ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖))
30		eliun 4928	. . . . . 6 ⊢ (𝑦 ∈ ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ↔ ∃𝑖 ∈ dom 𝑓 𝑦 ∈ (𝑓‘𝑖))
31	30	rexbii 3181	. . . . 5 ⊢ (∃𝑓 ∈ 𝐵 𝑦 ∈ ∪ 𝑖 ∈ dom 𝑓(𝑓‘𝑖) ↔ ∃𝑓 ∈ 𝐵 ∃𝑖 ∈ dom 𝑓 𝑦 ∈ (𝑓‘𝑖))
32	28, 29, 31	3bitri 297	. . . 4 ⊢ (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ ∃𝑓 ∈ 𝐵 ∃𝑖 ∈ dom 𝑓 𝑦 ∈ (𝑓‘𝑖))
33		df-rex 3070	. . . 4 ⊢ (∃𝑓 ∈ 𝐵 ∃𝑖 ∈ dom 𝑓 𝑦 ∈ (𝑓‘𝑖) ↔ ∃𝑓(𝑓 ∈ 𝐵 ∧ ∃𝑖 ∈ dom 𝑓 𝑦 ∈ (𝑓‘𝑖)))
34	26	abeq2i 2875	. . . . . 6 ⊢ (𝑓 ∈ 𝐵 ↔ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
35	34	anbi1i 624	. . . . 5 ⊢ ((𝑓 ∈ 𝐵 ∧ ∃𝑖 ∈ dom 𝑓 𝑦 ∈ (𝑓‘𝑖)) ↔ (∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ ∃𝑖 ∈ dom 𝑓 𝑦 ∈ (𝑓‘𝑖)))
36	35	exbii 1850	. . . 4 ⊢ (∃𝑓(𝑓 ∈ 𝐵 ∧ ∃𝑖 ∈ dom 𝑓 𝑦 ∈ (𝑓‘𝑖)) ↔ ∃𝑓(∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ ∃𝑖 ∈ dom 𝑓 𝑦 ∈ (𝑓‘𝑖)))
37	32, 33, 36	3bitri 297	. . 3 ⊢ (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ ∃𝑓(∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ∧ ∃𝑖 ∈ dom 𝑓 𝑦 ∈ (𝑓‘𝑖)))
38	17, 24, 37	3bitr4ri 304	. 2 ⊢ (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ ∃𝑓∃𝑛∃𝑖(𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑦 ∈ (𝑓‘𝑖)))
39		bnj643 32729	. . . . . 6 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑦 ∈ (𝑓‘𝑖)) → 𝜒)
40	18	bnj564 32724	. . . . . . 7 ⊢ (𝜒 → dom 𝑓 = 𝑛)
41	40	eleq2d 2824	. . . . . 6 ⊢ (𝜒 → (𝑖 ∈ dom 𝑓 ↔ 𝑖 ∈ 𝑛))
42		anbi1 632	. . . . . . 7 ⊢ ((𝑖 ∈ dom 𝑓 ↔ 𝑖 ∈ 𝑛) → ((𝑖 ∈ dom 𝑓 ∧ (𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑦 ∈ (𝑓‘𝑖))) ↔ (𝑖 ∈ 𝑛 ∧ (𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑦 ∈ (𝑓‘𝑖)))))
43		bnj334 32692	. . . . . . . 8 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑦 ∈ (𝑓‘𝑖)) ↔ (𝑖 ∈ dom 𝑓 ∧ 𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑦 ∈ (𝑓‘𝑖)))
44		bnj252 32682	. . . . . . . 8 ⊢ ((𝑖 ∈ dom 𝑓 ∧ 𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑦 ∈ (𝑓‘𝑖)) ↔ (𝑖 ∈ dom 𝑓 ∧ (𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑦 ∈ (𝑓‘𝑖))))
45	43, 44	bitri 274	. . . . . . 7 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑦 ∈ (𝑓‘𝑖)) ↔ (𝑖 ∈ dom 𝑓 ∧ (𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑦 ∈ (𝑓‘𝑖))))
46		bnj334 32692	. . . . . . . 8 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)) ↔ (𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑦 ∈ (𝑓‘𝑖)))
47		bnj252 32682	. . . . . . . 8 ⊢ ((𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑦 ∈ (𝑓‘𝑖)) ↔ (𝑖 ∈ 𝑛 ∧ (𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑦 ∈ (𝑓‘𝑖))))
48	46, 47	bitri 274	. . . . . . 7 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)) ↔ (𝑖 ∈ 𝑛 ∧ (𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑦 ∈ (𝑓‘𝑖))))
49	42, 45, 48	3bitr4g 314	. . . . . 6 ⊢ ((𝑖 ∈ dom 𝑓 ↔ 𝑖 ∈ 𝑛) → ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑦 ∈ (𝑓‘𝑖)) ↔ (𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖))))
50	39, 41, 49	3syl 18	. . . . 5 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑦 ∈ (𝑓‘𝑖)) → ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑦 ∈ (𝑓‘𝑖)) ↔ (𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖))))
51	50	ibi 266	. . . 4 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑦 ∈ (𝑓‘𝑖)) → (𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)))
52	51	2eximi 1838	. . 3 ⊢ (∃𝑛∃𝑖(𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑦 ∈ (𝑓‘𝑖)) → ∃𝑛∃𝑖(𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)))
53	52	eximi 1837	. 2 ⊢ (∃𝑓∃𝑛∃𝑖(𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑖 ∈ dom 𝑓 ∧ 𝑦 ∈ (𝑓‘𝑖)) → ∃𝑓∃𝑛∃𝑖(𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)))
54	38, 53	sylbi 216	1 ⊢ (𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓∃𝑛∃𝑖(𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (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:  bnj917  32914