Theorem relexpindlem 13651

Description: Principle of transitive induction, finite and non-class version. The first three hypotheses give various existences, the next three give necessary substitutions and the last two are the basis and the induction hypothesis. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.)

Hypotheses

Ref	Expression
relexpindlem.1	⊢ (𝜂 → Rel 𝑅)
relexpindlem.2	⊢ (𝜂 → 𝑅 ∈ V)
relexpindlem.3	⊢ (𝜂 → 𝑆 ∈ V)
relexpindlem.4	⊢ (𝑖 = 𝑆 → (𝜑 ↔ 𝜒))
relexpindlem.5	⊢ (𝑖 = 𝑥 → (𝜑 ↔ 𝜓))
relexpindlem.6	⊢ (𝑖 = 𝑗 → (𝜑 ↔ 𝜃))
relexpindlem.7	⊢ (𝜂 → 𝜒)
relexpindlem.8	⊢ (𝜂 → (𝑗𝑅𝑥 → (𝜃 → 𝜓)))

Assertion

Ref	Expression
relexpindlem	⊢ (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜓)))

Distinct variable groups:   𝑥,𝑖   𝑥,𝑛   𝑖,𝑗,𝑅,𝑥   𝑆,𝑖,𝑗,𝑥   𝜂,𝑖,𝑗,𝑥   𝜑,𝑗,𝑥   𝜓,𝑖,𝑗   𝜒,𝑖   𝜃,𝑖   𝜂,𝑥

Allowed substitution hints:   𝜑(𝑖,𝑛)   𝜓(𝑥,𝑛)   𝜒(𝑥,𝑗,𝑛)   𝜃(𝑥,𝑗,𝑛)   𝜂(𝑛)   𝑅(𝑛)   𝑆(𝑛)

Proof of Theorem relexpindlem

Dummy variables 𝑙 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 476	. . . 4 ⊢ ((𝜂 ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
2		eleq1 2676	. . . . . . 7 ⊢ (𝑘 = 0 → (𝑘 ∈ ℕ0 ↔ 0 ∈ ℕ0))
3	2	anbi2d 736	. . . . . 6 ⊢ (𝑘 = 0 → ((𝜂 ∧ 𝑘 ∈ ℕ0) ↔ (𝜂 ∧ 0 ∈ ℕ0)))
4		oveq2 6557	. . . . . . . . 9 ⊢ (𝑘 = 0 → (𝑅↑𝑟𝑘) = (𝑅↑𝑟0))
5	4	breqd 4594	. . . . . . . 8 ⊢ (𝑘 = 0 → (𝑆(𝑅↑𝑟𝑘)𝑥 ↔ 𝑆(𝑅↑𝑟0)𝑥))
6	5	imbi1d 330	. . . . . . 7 ⊢ (𝑘 = 0 → ((𝑆(𝑅↑𝑟𝑘)𝑥 → 𝜓) ↔ (𝑆(𝑅↑𝑟0)𝑥 → 𝜓)))
7	6	albidv 1836	. . . . . 6 ⊢ (𝑘 = 0 → (∀𝑥(𝑆(𝑅↑𝑟𝑘)𝑥 → 𝜓) ↔ ∀𝑥(𝑆(𝑅↑𝑟0)𝑥 → 𝜓)))
8	3, 7	imbi12d 333	. . . . 5 ⊢ (𝑘 = 0 → (((𝜂 ∧ 𝑘 ∈ ℕ0) → ∀𝑥(𝑆(𝑅↑𝑟𝑘)𝑥 → 𝜓)) ↔ ((𝜂 ∧ 0 ∈ ℕ0) → ∀𝑥(𝑆(𝑅↑𝑟0)𝑥 → 𝜓))))
9		eleq1 2676	. . . . . . 7 ⊢ (𝑘 = 𝑙 → (𝑘 ∈ ℕ0 ↔ 𝑙 ∈ ℕ0))
10	9	anbi2d 736	. . . . . 6 ⊢ (𝑘 = 𝑙 → ((𝜂 ∧ 𝑘 ∈ ℕ0) ↔ (𝜂 ∧ 𝑙 ∈ ℕ0)))
11		oveq2 6557	. . . . . . . . 9 ⊢ (𝑘 = 𝑙 → (𝑅↑𝑟𝑘) = (𝑅↑𝑟𝑙))
12	11	breqd 4594	. . . . . . . 8 ⊢ (𝑘 = 𝑙 → (𝑆(𝑅↑𝑟𝑘)𝑥 ↔ 𝑆(𝑅↑𝑟𝑙)𝑥))
13	12	imbi1d 330	. . . . . . 7 ⊢ (𝑘 = 𝑙 → ((𝑆(𝑅↑𝑟𝑘)𝑥 → 𝜓) ↔ (𝑆(𝑅↑𝑟𝑙)𝑥 → 𝜓)))
14	13	albidv 1836	. . . . . 6 ⊢ (𝑘 = 𝑙 → (∀𝑥(𝑆(𝑅↑𝑟𝑘)𝑥 → 𝜓) ↔ ∀𝑥(𝑆(𝑅↑𝑟𝑙)𝑥 → 𝜓)))
15	10, 14	imbi12d 333	. . . . 5 ⊢ (𝑘 = 𝑙 → (((𝜂 ∧ 𝑘 ∈ ℕ0) → ∀𝑥(𝑆(𝑅↑𝑟𝑘)𝑥 → 𝜓)) ↔ ((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑥(𝑆(𝑅↑𝑟𝑙)𝑥 → 𝜓))))
16		eleq1 2676	. . . . . . 7 ⊢ (𝑘 = (𝑙 + 1) → (𝑘 ∈ ℕ0 ↔ (𝑙 + 1) ∈ ℕ0))
17	16	anbi2d 736	. . . . . 6 ⊢ (𝑘 = (𝑙 + 1) → ((𝜂 ∧ 𝑘 ∈ ℕ0) ↔ (𝜂 ∧ (𝑙 + 1) ∈ ℕ0)))
18		oveq2 6557	. . . . . . . . 9 ⊢ (𝑘 = (𝑙 + 1) → (𝑅↑𝑟𝑘) = (𝑅↑𝑟(𝑙 + 1)))
19	18	breqd 4594	. . . . . . . 8 ⊢ (𝑘 = (𝑙 + 1) → (𝑆(𝑅↑𝑟𝑘)𝑥 ↔ 𝑆(𝑅↑𝑟(𝑙 + 1))𝑥))
20	19	imbi1d 330	. . . . . . 7 ⊢ (𝑘 = (𝑙 + 1) → ((𝑆(𝑅↑𝑟𝑘)𝑥 → 𝜓) ↔ (𝑆(𝑅↑𝑟(𝑙 + 1))𝑥 → 𝜓)))
21	20	albidv 1836	. . . . . 6 ⊢ (𝑘 = (𝑙 + 1) → (∀𝑥(𝑆(𝑅↑𝑟𝑘)𝑥 → 𝜓) ↔ ∀𝑥(𝑆(𝑅↑𝑟(𝑙 + 1))𝑥 → 𝜓)))
22	17, 21	imbi12d 333	. . . . 5 ⊢ (𝑘 = (𝑙 + 1) → (((𝜂 ∧ 𝑘 ∈ ℕ0) → ∀𝑥(𝑆(𝑅↑𝑟𝑘)𝑥 → 𝜓)) ↔ ((𝜂 ∧ (𝑙 + 1) ∈ ℕ0) → ∀𝑥(𝑆(𝑅↑𝑟(𝑙 + 1))𝑥 → 𝜓))))
23		eleq1 2676	. . . . . . 7 ⊢ (𝑘 = 𝑛 → (𝑘 ∈ ℕ0 ↔ 𝑛 ∈ ℕ0))
24	23	anbi2d 736	. . . . . 6 ⊢ (𝑘 = 𝑛 → ((𝜂 ∧ 𝑘 ∈ ℕ0) ↔ (𝜂 ∧ 𝑛 ∈ ℕ0)))
25		oveq2 6557	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → (𝑅↑𝑟𝑘) = (𝑅↑𝑟𝑛))
26	25	breqd 4594	. . . . . . . 8 ⊢ (𝑘 = 𝑛 → (𝑆(𝑅↑𝑟𝑘)𝑥 ↔ 𝑆(𝑅↑𝑟𝑛)𝑥))
27	26	imbi1d 330	. . . . . . 7 ⊢ (𝑘 = 𝑛 → ((𝑆(𝑅↑𝑟𝑘)𝑥 → 𝜓) ↔ (𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜓)))
28	27	albidv 1836	. . . . . 6 ⊢ (𝑘 = 𝑛 → (∀𝑥(𝑆(𝑅↑𝑟𝑘)𝑥 → 𝜓) ↔ ∀𝑥(𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜓)))
29	24, 28	imbi12d 333	. . . . 5 ⊢ (𝑘 = 𝑛 → (((𝜂 ∧ 𝑘 ∈ ℕ0) → ∀𝑥(𝑆(𝑅↑𝑟𝑘)𝑥 → 𝜓)) ↔ ((𝜂 ∧ 𝑛 ∈ ℕ0) → ∀𝑥(𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜓))))
30		relexpindlem.2	. . . . . . . . . 10 ⊢ (𝜂 → 𝑅 ∈ V)
31		relexpindlem.1	. . . . . . . . . 10 ⊢ (𝜂 → Rel 𝑅)
32	30, 31	jca 553	. . . . . . . . 9 ⊢ (𝜂 → (𝑅 ∈ V ∧ Rel 𝑅))
33	32	adantr 480	. . . . . . . 8 ⊢ ((𝜂 ∧ 0 ∈ ℕ0) → (𝑅 ∈ V ∧ Rel 𝑅))
34		relexp0 13611	. . . . . . . 8 ⊢ ((𝑅 ∈ V ∧ Rel 𝑅) → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅))
35	33, 34	syl 17	. . . . . . 7 ⊢ ((𝜂 ∧ 0 ∈ ℕ0) → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅))
36		relexpindlem.7	. . . . . . . . . . 11 ⊢ (𝜂 → 𝜒)
37	36	adantr 480	. . . . . . . . . 10 ⊢ ((𝜂 ∧ 0 ∈ ℕ0) → 𝜒)
38		simpl 472	. . . . . . . . . 10 ⊢ ((𝜂 ∧ 0 ∈ ℕ0) → 𝜂)
39		relexpindlem.3	. . . . . . . . . . . 12 ⊢ (𝜂 → 𝑆 ∈ V)
40	39	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜒 ∧ 𝜂) → 𝑆 ∈ V)
41		simprl 790	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 = 𝑆 ∧ (𝜂 ∧ (𝜑 ∧ 𝑖 = 𝑆))) → 𝜂)
42	41, 36	jccil 561	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 = 𝑆 ∧ (𝜂 ∧ (𝜑 ∧ 𝑖 = 𝑆))) → (𝜒 ∧ 𝜂))
43	42	expcom 450	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜂 ∧ (𝜑 ∧ 𝑖 = 𝑆)) → (𝑖 = 𝑆 → (𝜒 ∧ 𝜂)))
44	43	expcom 450	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 = 𝑆) → (𝜂 → (𝑖 = 𝑆 → (𝜒 ∧ 𝜂))))
45	44	expcom 450	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑆 → (𝜑 → (𝜂 → (𝑖 = 𝑆 → (𝜒 ∧ 𝜂)))))
46	45	3imp1 1272	. . . . . . . . . . . . . 14 ⊢ (((𝑖 = 𝑆 ∧ 𝜑 ∧ 𝜂) ∧ 𝑖 = 𝑆) → (𝜒 ∧ 𝜂))
47	46	expcom 450	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑆 → ((𝑖 = 𝑆 ∧ 𝜑 ∧ 𝜂) → (𝜒 ∧ 𝜂)))
48		simpr 476	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜂 ∧ 𝑖 = 𝑆) → 𝑖 = 𝑆)
49	48	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜒 ∧ (𝜂 ∧ 𝑖 = 𝑆)) → 𝑖 = 𝑆)
50		relexpindlem.4	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑆 → (𝜑 ↔ 𝜒))
51	50	ad2antll 761	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜒 ∧ (𝜂 ∧ 𝑖 = 𝑆)) → (𝜑 ↔ 𝜒))
52	51	bicomd 212	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜒 ∧ (𝜂 ∧ 𝑖 = 𝑆)) → (𝜒 ↔ 𝜑))
53		anbi1 739	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜒 ↔ 𝜑) → ((𝜒 ∧ (𝜂 ∧ 𝑖 = 𝑆)) ↔ (𝜑 ∧ (𝜂 ∧ 𝑖 = 𝑆))))
54		simpl 472	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝜂 ∧ 𝑖 = 𝑆)) → 𝜑)
55	53, 54	syl6bi 242	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜒 ↔ 𝜑) → ((𝜒 ∧ (𝜂 ∧ 𝑖 = 𝑆)) → 𝜑))
56	52, 55	mpcom 37	. . . . . . . . . . . . . . . 16 ⊢ ((𝜒 ∧ (𝜂 ∧ 𝑖 = 𝑆)) → 𝜑)
57		simprl 790	. . . . . . . . . . . . . . . 16 ⊢ ((𝜒 ∧ (𝜂 ∧ 𝑖 = 𝑆)) → 𝜂)
58	49, 56, 57	3jca 1235	. . . . . . . . . . . . . . 15 ⊢ ((𝜒 ∧ (𝜂 ∧ 𝑖 = 𝑆)) → (𝑖 = 𝑆 ∧ 𝜑 ∧ 𝜂))
59	58	anassrs 678	. . . . . . . . . . . . . 14 ⊢ (((𝜒 ∧ 𝜂) ∧ 𝑖 = 𝑆) → (𝑖 = 𝑆 ∧ 𝜑 ∧ 𝜂))
60	59	expcom 450	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑆 → ((𝜒 ∧ 𝜂) → (𝑖 = 𝑆 ∧ 𝜑 ∧ 𝜂)))
61	47, 60	impbid 201	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑆 → ((𝑖 = 𝑆 ∧ 𝜑 ∧ 𝜂) ↔ (𝜒 ∧ 𝜂)))
62	61	spcegv 3267	. . . . . . . . . . 11 ⊢ (𝑆 ∈ V → ((𝜒 ∧ 𝜂) → ∃𝑖(𝑖 = 𝑆 ∧ 𝜑 ∧ 𝜂)))
63	40, 62	mpcom 37	. . . . . . . . . 10 ⊢ ((𝜒 ∧ 𝜂) → ∃𝑖(𝑖 = 𝑆 ∧ 𝜑 ∧ 𝜂))
64	37, 38, 63	syl2anc 691	. . . . . . . . 9 ⊢ ((𝜂 ∧ 0 ∈ ℕ0) → ∃𝑖(𝑖 = 𝑆 ∧ 𝜑 ∧ 𝜂))
65		simpl 472	. . . . . . . . . . . . . 14 ⊢ ((𝑆( I ↾ ∪ ∪ 𝑅)𝑥 ∧ (∃𝑖(𝑖 = 𝑆 ∧ 𝜑 ∧ 𝜂) ∧ (𝜂 ∧ 0 ∈ ℕ0))) → 𝑆( I ↾ ∪ ∪ 𝑅)𝑥)
66		df-br 4584	. . . . . . . . . . . . . 14 ⊢ (𝑆( I ↾ ∪ ∪ 𝑅)𝑥 ↔ ⟨𝑆, 𝑥⟩ ∈ ( I ↾ ∪ ∪ 𝑅))
67	65, 66	sylib 207	. . . . . . . . . . . . 13 ⊢ ((𝑆( I ↾ ∪ ∪ 𝑅)𝑥 ∧ (∃𝑖(𝑖 = 𝑆 ∧ 𝜑 ∧ 𝜂) ∧ (𝜂 ∧ 0 ∈ ℕ0))) → ⟨𝑆, 𝑥⟩ ∈ ( I ↾ ∪ ∪ 𝑅))
68		vex 3176	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
69	68	opelres 5322	. . . . . . . . . . . . 13 ⊢ (⟨𝑆, 𝑥⟩ ∈ ( I ↾ ∪ ∪ 𝑅) ↔ (⟨𝑆, 𝑥⟩ ∈ I ∧ 𝑆 ∈ ∪ ∪ 𝑅))
70	67, 69	sylib 207	. . . . . . . . . . . 12 ⊢ ((𝑆( I ↾ ∪ ∪ 𝑅)𝑥 ∧ (∃𝑖(𝑖 = 𝑆 ∧ 𝜑 ∧ 𝜂) ∧ (𝜂 ∧ 0 ∈ ℕ0))) → (⟨𝑆, 𝑥⟩ ∈ I ∧ 𝑆 ∈ ∪ ∪ 𝑅))
71		simpll 786	. . . . . . . . . . . . . 14 ⊢ (((⟨𝑆, 𝑥⟩ ∈ I ∧ 𝑆 ∈ ∪ ∪ 𝑅) ∧ (𝑆( I ↾ ∪ ∪ 𝑅)𝑥 ∧ (∃𝑖(𝑖 = 𝑆 ∧ 𝜑 ∧ 𝜂) ∧ (𝜂 ∧ 0 ∈ ℕ0)))) → ⟨𝑆, 𝑥⟩ ∈ I )
72		df-br 4584	. . . . . . . . . . . . . 14 ⊢ (𝑆 I 𝑥 ↔ ⟨𝑆, 𝑥⟩ ∈ I )
73	71, 72	sylibr 223	. . . . . . . . . . . . 13 ⊢ (((⟨𝑆, 𝑥⟩ ∈ I ∧ 𝑆 ∈ ∪ ∪ 𝑅) ∧ (𝑆( I ↾ ∪ ∪ 𝑅)𝑥 ∧ (∃𝑖(𝑖 = 𝑆 ∧ 𝜑 ∧ 𝜂) ∧ (𝜂 ∧ 0 ∈ ℕ0)))) → 𝑆 I 𝑥)
74	68	ideq 5196	. . . . . . . . . . . . 13 ⊢ (𝑆 I 𝑥 ↔ 𝑆 = 𝑥)
75	73, 74	sylib 207	. . . . . . . . . . . 12 ⊢ (((⟨𝑆, 𝑥⟩ ∈ I ∧ 𝑆 ∈ ∪ ∪ 𝑅) ∧ (𝑆( I ↾ ∪ ∪ 𝑅)𝑥 ∧ (∃𝑖(𝑖 = 𝑆 ∧ 𝜑 ∧ 𝜂) ∧ (𝜂 ∧ 0 ∈ ℕ0)))) → 𝑆 = 𝑥)
76	70, 75	mpancom 700	. . . . . . . . . . 11 ⊢ ((𝑆( I ↾ ∪ ∪ 𝑅)𝑥 ∧ (∃𝑖(𝑖 = 𝑆 ∧ 𝜑 ∧ 𝜂) ∧ (𝜂 ∧ 0 ∈ ℕ0))) → 𝑆 = 𝑥)
77		breq1 4586	. . . . . . . . . . . . 13 ⊢ (𝑆 = 𝑥 → (𝑆( I ↾ ∪ ∪ 𝑅)𝑥 ↔ 𝑥( I ↾ ∪ ∪ 𝑅)𝑥))
78		eqeq2 2621	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 = 𝑥 → (𝑖 = 𝑆 ↔ 𝑖 = 𝑥))
79	78	3anbi1d 1395	. . . . . . . . . . . . . . 15 ⊢ (𝑆 = 𝑥 → ((𝑖 = 𝑆 ∧ 𝜑 ∧ 𝜂) ↔ (𝑖 = 𝑥 ∧ 𝜑 ∧ 𝜂)))
80	79	exbidv 1837	. . . . . . . . . . . . . 14 ⊢ (𝑆 = 𝑥 → (∃𝑖(𝑖 = 𝑆 ∧ 𝜑 ∧ 𝜂) ↔ ∃𝑖(𝑖 = 𝑥 ∧ 𝜑 ∧ 𝜂)))
81	80	anbi1d 737	. . . . . . . . . . . . 13 ⊢ (𝑆 = 𝑥 → ((∃𝑖(𝑖 = 𝑆 ∧ 𝜑 ∧ 𝜂) ∧ (𝜂 ∧ 0 ∈ ℕ0)) ↔ (∃𝑖(𝑖 = 𝑥 ∧ 𝜑 ∧ 𝜂) ∧ (𝜂 ∧ 0 ∈ ℕ0))))
82	77, 81	anbi12d 743	. . . . . . . . . . . 12 ⊢ (𝑆 = 𝑥 → ((𝑆( I ↾ ∪ ∪ 𝑅)𝑥 ∧ (∃𝑖(𝑖 = 𝑆 ∧ 𝜑 ∧ 𝜂) ∧ (𝜂 ∧ 0 ∈ ℕ0))) ↔ (𝑥( I ↾ ∪ ∪ 𝑅)𝑥 ∧ (∃𝑖(𝑖 = 𝑥 ∧ 𝜑 ∧ 𝜂) ∧ (𝜂 ∧ 0 ∈ ℕ0)))))
83		simprl 790	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜂 ∧ (𝜑 ∧ 𝑖 = 𝑥)) → 𝜑)
84		relexpindlem.5	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 = 𝑥 → (𝜑 ↔ 𝜓))
85	84	ad2antll 761	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜂 ∧ (𝜑 ∧ 𝑖 = 𝑥)) → (𝜑 ↔ 𝜓))
86	83, 85	mpbid 221	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜂 ∧ (𝜑 ∧ 𝑖 = 𝑥)) → 𝜓)
87	86	expcom 450	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 = 𝑥) → (𝜂 → 𝜓))
88	87	expcom 450	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑥 → (𝜑 → (𝜂 → 𝜓)))
89	88	3imp 1249	. . . . . . . . . . . . . 14 ⊢ ((𝑖 = 𝑥 ∧ 𝜑 ∧ 𝜂) → 𝜓)
90	89	exlimiv 1845	. . . . . . . . . . . . 13 ⊢ (∃𝑖(𝑖 = 𝑥 ∧ 𝜑 ∧ 𝜂) → 𝜓)
91	90	ad2antrl 760	. . . . . . . . . . . 12 ⊢ ((𝑥( I ↾ ∪ ∪ 𝑅)𝑥 ∧ (∃𝑖(𝑖 = 𝑥 ∧ 𝜑 ∧ 𝜂) ∧ (𝜂 ∧ 0 ∈ ℕ0))) → 𝜓)
92	82, 91	syl6bi 242	. . . . . . . . . . 11 ⊢ (𝑆 = 𝑥 → ((𝑆( I ↾ ∪ ∪ 𝑅)𝑥 ∧ (∃𝑖(𝑖 = 𝑆 ∧ 𝜑 ∧ 𝜂) ∧ (𝜂 ∧ 0 ∈ ℕ0))) → 𝜓))
93	76, 92	mpcom 37	. . . . . . . . . 10 ⊢ ((𝑆( I ↾ ∪ ∪ 𝑅)𝑥 ∧ (∃𝑖(𝑖 = 𝑆 ∧ 𝜑 ∧ 𝜂) ∧ (𝜂 ∧ 0 ∈ ℕ0))) → 𝜓)
94	93	expcom 450	. . . . . . . . 9 ⊢ ((∃𝑖(𝑖 = 𝑆 ∧ 𝜑 ∧ 𝜂) ∧ (𝜂 ∧ 0 ∈ ℕ0)) → (𝑆( I ↾ ∪ ∪ 𝑅)𝑥 → 𝜓))
95	64, 94	mpancom 700	. . . . . . . 8 ⊢ ((𝜂 ∧ 0 ∈ ℕ0) → (𝑆( I ↾ ∪ ∪ 𝑅)𝑥 → 𝜓))
96		breq 4585	. . . . . . . . 9 ⊢ ((𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅) → (𝑆(𝑅↑𝑟0)𝑥 ↔ 𝑆( I ↾ ∪ ∪ 𝑅)𝑥))
97	96	imbi1d 330	. . . . . . . 8 ⊢ ((𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅) → ((𝑆(𝑅↑𝑟0)𝑥 → 𝜓) ↔ (𝑆( I ↾ ∪ ∪ 𝑅)𝑥 → 𝜓)))
98	95, 97	syl5ibr 235	. . . . . . 7 ⊢ ((𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅) → ((𝜂 ∧ 0 ∈ ℕ0) → (𝑆(𝑅↑𝑟0)𝑥 → 𝜓)))
99	35, 98	mpcom 37	. . . . . 6 ⊢ ((𝜂 ∧ 0 ∈ ℕ0) → (𝑆(𝑅↑𝑟0)𝑥 → 𝜓))
100	99	alrimiv 1842	. . . . 5 ⊢ ((𝜂 ∧ 0 ∈ ℕ0) → ∀𝑥(𝑆(𝑅↑𝑟0)𝑥 → 𝜓))
101		breq2 4587	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑥 → (𝑆(𝑅↑𝑟𝑙)𝑖 ↔ 𝑆(𝑅↑𝑟𝑙)𝑥))
102	101, 84	imbi12d 333	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑥 → ((𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑) ↔ (𝑆(𝑅↑𝑟𝑙)𝑥 → 𝜓)))
103	102	cbvalv 2261	. . . . . . . . . 10 ⊢ (∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑) ↔ ∀𝑥(𝑆(𝑅↑𝑟𝑙)𝑥 → 𝜓))
104	103	bicomi 213	. . . . . . . . 9 ⊢ (∀𝑥(𝑆(𝑅↑𝑟𝑙)𝑥 → 𝜓) ↔ ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑))
105		imbi2 337	. . . . . . . . . . . . 13 ⊢ ((∀𝑥(𝑆(𝑅↑𝑟𝑙)𝑥 → 𝜓) ↔ ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) → (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑥(𝑆(𝑅↑𝑟𝑙)𝑥 → 𝜓)) ↔ ((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑))))
106	105	anbi1d 737	. . . . . . . . . . . 12 ⊢ ((∀𝑥(𝑆(𝑅↑𝑟𝑙)𝑥 → 𝜓) ↔ ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) → ((((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑥(𝑆(𝑅↑𝑟𝑙)𝑥 → 𝜓)) ∧ 𝑙 ∈ ℕ0) ↔ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ 𝑙 ∈ ℕ0)))
107	106	anbi2d 736	. . . . . . . . . . 11 ⊢ ((∀𝑥(𝑆(𝑅↑𝑟𝑙)𝑥 → 𝜓) ↔ ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) → (((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑥(𝑆(𝑅↑𝑟𝑙)𝑥 → 𝜓)) ∧ 𝑙 ∈ ℕ0)) ↔ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ 𝑙 ∈ ℕ0))))
108	107	anbi2d 736	. . . . . . . . . 10 ⊢ ((∀𝑥(𝑆(𝑅↑𝑟𝑙)𝑥 → 𝜓) ↔ ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) → ((𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑥(𝑆(𝑅↑𝑟𝑙)𝑥 → 𝜓)) ∧ 𝑙 ∈ ℕ0))) ↔ (𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ 𝑙 ∈ ℕ0)))))
109	30	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ 𝑙 ∈ ℕ0))) → 𝑅 ∈ V)
110	31	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ 𝑙 ∈ ℕ0))) → Rel 𝑅)
111		simprrr 801	. . . . . . . . . . . . 13 ⊢ ((𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ 𝑙 ∈ ℕ0))) → 𝑙 ∈ ℕ0)
112		relexpsucl 13621	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ V ∧ Rel 𝑅 ∧ 𝑙 ∈ ℕ0) → (𝑅↑𝑟(𝑙 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑙)))
113	109, 110, 111, 112	syl3anc 1318	. . . . . . . . . . . 12 ⊢ ((𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ 𝑙 ∈ ℕ0))) → (𝑅↑𝑟(𝑙 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑙)))
114		simpl 472	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 ∧ (𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ 𝑙 ∈ ℕ0)))) → 𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥)
115	39	ad2antrl 760	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 ∧ (𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ 𝑙 ∈ ℕ0)))) → 𝑆 ∈ V)
116		brcog 5210	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ∈ V ∧ 𝑥 ∈ V) → (𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 ↔ ∃𝑗(𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥)))
117	115, 68, 116	sylancl 693	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 ∧ (𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ 𝑙 ∈ ℕ0)))) → (𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 ↔ ∃𝑗(𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥)))
118	114, 117	mpbid 221	. . . . . . . . . . . . . . 15 ⊢ ((𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 ∧ (𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ 𝑙 ∈ ℕ0)))) → ∃𝑗(𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥))
119		simprl 790	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 ∧ (𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ (𝑙 ∈ ℕ0 ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥)))))) → 𝜂)
120		simprrl 800	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ (𝑙 ∈ ℕ0 ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥)))) → 𝑙 ∈ ℕ0)
121	120	ad2antll 761	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 ∧ (𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ (𝑙 ∈ ℕ0 ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥)))))) → 𝑙 ∈ ℕ0)
122	119, 121	jca 553	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 ∧ (𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ (𝑙 ∈ ℕ0 ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥)))))) → (𝜂 ∧ 𝑙 ∈ ℕ0))
123		simprl 790	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ (𝑙 ∈ ℕ0 ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥)))) → ((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)))
124	123	ad2antll 761	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 ∧ (𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ (𝑙 ∈ ℕ0 ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥)))))) → ((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)))
125	122, 124	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 ∧ (𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ (𝑙 ∈ ℕ0 ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥)))))) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑))
126		simprrl 800	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑) ∧ (𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 ∧ (𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ (𝑙 ∈ ℕ0 ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥))))))) → 𝜂)
127		simprrr 801	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ (𝑙 ∈ ℕ0 ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥))) → 𝑗𝑅𝑥)
128	127	ad2antll 761	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ (𝑙 ∈ ℕ0 ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥))))) → 𝑗𝑅𝑥)
129	128	ad2antll 761	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑) ∧ (𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 ∧ (𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ (𝑙 ∈ ℕ0 ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥))))))) → 𝑗𝑅𝑥)
130		breq2 4587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑖 = 𝑗 → (𝑆(𝑅↑𝑟𝑙)𝑖 ↔ 𝑆(𝑅↑𝑟𝑙)𝑗))
131		relexpindlem.6	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑖 = 𝑗 → (𝜑 ↔ 𝜃))
132	130, 131	imbi12d 333	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑖 = 𝑗 → ((𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑) ↔ (𝑆(𝑅↑𝑟𝑙)𝑗 → 𝜃)))
133	132	cbvalv 2261	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑) ↔ ∀𝑗(𝑆(𝑅↑𝑟𝑙)𝑗 → 𝜃))
134		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑) ↔ ∀𝑗(𝑆(𝑅↑𝑟𝑙)𝑗 → 𝜃)) → (∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑) ↔ ∀𝑗(𝑆(𝑅↑𝑟𝑙)𝑗 → 𝜃)))
135		imbi2 337	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑) ↔ ∀𝑗(𝑆(𝑅↑𝑟𝑙)𝑗 → 𝜃)) → (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ↔ ((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑗(𝑆(𝑅↑𝑟𝑙)𝑗 → 𝜃))))
136	135	anbi1d 737	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑) ↔ ∀𝑗(𝑆(𝑅↑𝑟𝑙)𝑗 → 𝜃)) → ((((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ (𝑙 ∈ ℕ0 ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥))) ↔ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑗(𝑆(𝑅↑𝑟𝑙)𝑗 → 𝜃)) ∧ (𝑙 ∈ ℕ0 ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥)))))
137	136	anbi2d 736	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑) ↔ ∀𝑗(𝑆(𝑅↑𝑟𝑙)𝑗 → 𝜃)) → (((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ (𝑙 ∈ ℕ0 ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥)))) ↔ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑗(𝑆(𝑅↑𝑟𝑙)𝑗 → 𝜃)) ∧ (𝑙 ∈ ℕ0 ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥))))))
138	137	anbi2d 736	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑) ↔ ∀𝑗(𝑆(𝑅↑𝑟𝑙)𝑗 → 𝜃)) → ((𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ (𝑙 ∈ ℕ0 ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥))))) ↔ (𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑗(𝑆(𝑅↑𝑟𝑙)𝑗 → 𝜃)) ∧ (𝑙 ∈ ℕ0 ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥)))))))
139	138	anbi2d 736	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑) ↔ ∀𝑗(𝑆(𝑅↑𝑟𝑙)𝑗 → 𝜃)) → ((𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 ∧ (𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ (𝑙 ∈ ℕ0 ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥)))))) ↔ (𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 ∧ (𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑗(𝑆(𝑅↑𝑟𝑙)𝑗 → 𝜃)) ∧ (𝑙 ∈ ℕ0 ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥))))))))
140	134, 139	anbi12d 743	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑) ↔ ∀𝑗(𝑆(𝑅↑𝑟𝑙)𝑗 → 𝜃)) → ((∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑) ∧ (𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 ∧ (𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ (𝑙 ∈ ℕ0 ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥))))))) ↔ (∀𝑗(𝑆(𝑅↑𝑟𝑙)𝑗 → 𝜃) ∧ (𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 ∧ (𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑗(𝑆(𝑅↑𝑟𝑙)𝑗 → 𝜃)) ∧ (𝑙 ∈ ℕ0 ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥)))))))))
141		simprrl 800	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑗(𝑆(𝑅↑𝑟𝑙)𝑗 → 𝜃)) ∧ (𝑙 ∈ ℕ0 ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥))) → 𝑆(𝑅↑𝑟𝑙)𝑗)
142	141	ad2antll 761	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑗(𝑆(𝑅↑𝑟𝑙)𝑗 → 𝜃)) ∧ (𝑙 ∈ ℕ0 ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥))))) → 𝑆(𝑅↑𝑟𝑙)𝑗)
143	142	ad2antll 761	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((∀𝑗(𝑆(𝑅↑𝑟𝑙)𝑗 → 𝜃) ∧ (𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 ∧ (𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑗(𝑆(𝑅↑𝑟𝑙)𝑗 → 𝜃)) ∧ (𝑙 ∈ ℕ0 ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥))))))) → 𝑆(𝑅↑𝑟𝑙)𝑗)
144		sp 2041	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (∀𝑗(𝑆(𝑅↑𝑟𝑙)𝑗 → 𝜃) → (𝑆(𝑅↑𝑟𝑙)𝑗 → 𝜃))
145	144	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((∀𝑗(𝑆(𝑅↑𝑟𝑙)𝑗 → 𝜃) ∧ (𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 ∧ (𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑗(𝑆(𝑅↑𝑟𝑙)𝑗 → 𝜃)) ∧ (𝑙 ∈ ℕ0 ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥))))))) → (𝑆(𝑅↑𝑟𝑙)𝑗 → 𝜃))
146	143, 145	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((∀𝑗(𝑆(𝑅↑𝑟𝑙)𝑗 → 𝜃) ∧ (𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 ∧ (𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑗(𝑆(𝑅↑𝑟𝑙)𝑗 → 𝜃)) ∧ (𝑙 ∈ ℕ0 ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥))))))) → 𝜃)
147	140, 146	syl6bi 242	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑) ↔ ∀𝑗(𝑆(𝑅↑𝑟𝑙)𝑗 → 𝜃)) → ((∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑) ∧ (𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 ∧ (𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ (𝑙 ∈ ℕ0 ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥))))))) → 𝜃))
148	133, 147	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑) ∧ (𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 ∧ (𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ (𝑙 ∈ ℕ0 ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥))))))) → 𝜃)
149		relexpindlem.8	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜂 → (𝑗𝑅𝑥 → (𝜃 → 𝜓)))
150	126, 129, 148, 149	syl3c 64	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑) ∧ (𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 ∧ (𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ (𝑙 ∈ ℕ0 ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥))))))) → 𝜓)
151	125, 150	mpancom 700	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 ∧ (𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ (𝑙 ∈ ℕ0 ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥)))))) → 𝜓)
152	151	expcom 450	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ (𝑙 ∈ ℕ0 ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥))))) → (𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 → 𝜓))
153	152	expcom 450	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ (𝑙 ∈ ℕ0 ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥)))) → (𝜂 → (𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 → 𝜓)))
154	153	expcom 450	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ (𝑙 ∈ ℕ0 ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥))) → ((𝑙 + 1) ∈ ℕ0 → (𝜂 → (𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 → 𝜓))))
155	154	anassrs 678	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ 𝑙 ∈ ℕ0) ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥)) → ((𝑙 + 1) ∈ ℕ0 → (𝜂 → (𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 → 𝜓))))
156	155	impcom 445	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑙 + 1) ∈ ℕ0 ∧ ((((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ 𝑙 ∈ ℕ0) ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥))) → (𝜂 → (𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 → 𝜓)))
157	156	anassrs 678	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ 𝑙 ∈ ℕ0)) ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥)) → (𝜂 → (𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 → 𝜓)))
158	157	impcom 445	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜂 ∧ (((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ 𝑙 ∈ ℕ0)) ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥))) → (𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 → 𝜓))
159	158	anassrs 678	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ 𝑙 ∈ ℕ0))) ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥)) → (𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 → 𝜓))
160	159	impcom 445	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 ∧ ((𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ 𝑙 ∈ ℕ0))) ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥))) → 𝜓)
161	160	anassrs 678	. . . . . . . . . . . . . . 15 ⊢ (((𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 ∧ (𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ 𝑙 ∈ ℕ0)))) ∧ (𝑆(𝑅↑𝑟𝑙)𝑗 ∧ 𝑗𝑅𝑥)) → 𝜓)
162	118, 161	exlimddv 1850	. . . . . . . . . . . . . 14 ⊢ ((𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 ∧ (𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ 𝑙 ∈ ℕ0)))) → 𝜓)
163	162	expcom 450	. . . . . . . . . . . . 13 ⊢ ((𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ 𝑙 ∈ ℕ0))) → (𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 → 𝜓))
164		breq 4585	. . . . . . . . . . . . . 14 ⊢ ((𝑅↑𝑟(𝑙 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑙)) → (𝑆(𝑅↑𝑟(𝑙 + 1))𝑥 ↔ 𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥))
165	164	imbi1d 330	. . . . . . . . . . . . 13 ⊢ ((𝑅↑𝑟(𝑙 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑙)) → ((𝑆(𝑅↑𝑟(𝑙 + 1))𝑥 → 𝜓) ↔ (𝑆(𝑅 ∘ (𝑅↑𝑟𝑙))𝑥 → 𝜓)))
166	163, 165	syl5ibr 235	. . . . . . . . . . . 12 ⊢ ((𝑅↑𝑟(𝑙 + 1)) = (𝑅 ∘ (𝑅↑𝑟𝑙)) → ((𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ 𝑙 ∈ ℕ0))) → (𝑆(𝑅↑𝑟(𝑙 + 1))𝑥 → 𝜓)))
167	113, 166	mpcom 37	. . . . . . . . . . 11 ⊢ ((𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ 𝑙 ∈ ℕ0))) → (𝑆(𝑅↑𝑟(𝑙 + 1))𝑥 → 𝜓))
168	167	alrimiv 1842	. . . . . . . . . 10 ⊢ ((𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) ∧ 𝑙 ∈ ℕ0))) → ∀𝑥(𝑆(𝑅↑𝑟(𝑙 + 1))𝑥 → 𝜓))
169	108, 168	syl6bi 242	. . . . . . . . 9 ⊢ ((∀𝑥(𝑆(𝑅↑𝑟𝑙)𝑥 → 𝜓) ↔ ∀𝑖(𝑆(𝑅↑𝑟𝑙)𝑖 → 𝜑)) → ((𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑥(𝑆(𝑅↑𝑟𝑙)𝑥 → 𝜓)) ∧ 𝑙 ∈ ℕ0))) → ∀𝑥(𝑆(𝑅↑𝑟(𝑙 + 1))𝑥 → 𝜓)))
170	104, 169	ax-mp 5	. . . . . . . 8 ⊢ ((𝜂 ∧ ((𝑙 + 1) ∈ ℕ0 ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑥(𝑆(𝑅↑𝑟𝑙)𝑥 → 𝜓)) ∧ 𝑙 ∈ ℕ0))) → ∀𝑥(𝑆(𝑅↑𝑟(𝑙 + 1))𝑥 → 𝜓))
171	170	anassrs 678	. . . . . . 7 ⊢ (((𝜂 ∧ (𝑙 + 1) ∈ ℕ0) ∧ (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑥(𝑆(𝑅↑𝑟𝑙)𝑥 → 𝜓)) ∧ 𝑙 ∈ ℕ0)) → ∀𝑥(𝑆(𝑅↑𝑟(𝑙 + 1))𝑥 → 𝜓))
172	171	expcom 450	. . . . . 6 ⊢ ((((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑥(𝑆(𝑅↑𝑟𝑙)𝑥 → 𝜓)) ∧ 𝑙 ∈ ℕ0) → ((𝜂 ∧ (𝑙 + 1) ∈ ℕ0) → ∀𝑥(𝑆(𝑅↑𝑟(𝑙 + 1))𝑥 → 𝜓)))
173	172	expcom 450	. . . . 5 ⊢ (𝑙 ∈ ℕ0 → (((𝜂 ∧ 𝑙 ∈ ℕ0) → ∀𝑥(𝑆(𝑅↑𝑟𝑙)𝑥 → 𝜓)) → ((𝜂 ∧ (𝑙 + 1) ∈ ℕ0) → ∀𝑥(𝑆(𝑅↑𝑟(𝑙 + 1))𝑥 → 𝜓))))
174	8, 15, 22, 29, 100, 173	nn0ind 11348	. . . 4 ⊢ (𝑛 ∈ ℕ0 → ((𝜂 ∧ 𝑛 ∈ ℕ0) → ∀𝑥(𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜓)))
175	1, 174	mpcom 37	. . 3 ⊢ ((𝜂 ∧ 𝑛 ∈ ℕ0) → ∀𝑥(𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜓))
176	175	19.21bi 2047	. 2 ⊢ ((𝜂 ∧ 𝑛 ∈ ℕ0) → (𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜓))
177	176	ex 449	1 ⊢ (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜓)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131  ∪ cuni 4372   class class class wbr 4583   I cid 4948   ↾ cres 5040   ∘ ccom 5042  Rel wrel 5043  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  ℕ₀cn0 11169  ↑_𝑟crelexp 13608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-seq 12664  df-relexp 13609

This theorem is referenced by:  relexpind  13652