Theorem fseqenlem2 10028

Description: Lemma for fseqen 10030. (Contributed by Mario Carneiro, 17-May-2015.)

Hypotheses

Ref	Expression
fseqenlem.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
fseqenlem.b	⊢ (𝜑 → 𝐵 ∈ 𝐴)
fseqenlem.f	⊢ (𝜑 → 𝐹:(𝐴 × 𝐴)–1-1-onto→𝐴)
fseqenlem.g	⊢ 𝐺 = seqω((𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴 ↑m suc 𝑛) ↦ ((𝑓‘(𝑥 ↾ 𝑛))𝐹(𝑥‘𝑛)))), {⟨∅, 𝐵⟩})
fseqenlem.k	⊢ 𝐾 = (𝑦 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘) ↦ ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩)

Assertion

Ref	Expression
fseqenlem2	⊢ (𝜑 → 𝐾:∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘)–1-1→(ω × 𝐴))

Distinct variable groups:   𝑦,𝐵   𝑓,𝑛,𝑥,𝐹   𝑦,𝑘,𝐺   𝑓,𝑘,𝑦,𝐴,𝑛,𝑥   𝜑,𝑘,𝑛,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑥, 𝑓, 𝑘, 𝑛)   𝐹(𝑦, 𝑘)   𝐺(𝑥, 𝑓, 𝑛)   𝐾(𝑥, 𝑦, 𝑓, 𝑘, 𝑛)   𝑉(𝑥, 𝑦, 𝑓, 𝑘, 𝑛)

Proof of Theorem fseqenlem2

Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliun 4965	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘) ↔ ∃𝑘 ∈ ω 𝑦 ∈ (𝐴 ↑m 𝑘))
2		elmapi 8855	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐴 ↑m 𝑘) → 𝑦:𝑘⟶𝐴)
3	2	ad2antll 742	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑m 𝑘))) → 𝑦:𝑘⟶𝐴)
4	3	fdmd 6723	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑m 𝑘))) → dom 𝑦 = 𝑘)
5		simprl 783	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑m 𝑘))) → 𝑘 ∈ ω)
6	4, 5	eqeltrd 2866	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑m 𝑘))) → dom 𝑦 ∈ ω)
7	4	fveq2d 6892	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑m 𝑘))) → (𝐺‘dom 𝑦) = (𝐺‘𝑘))
8	7	fveq1d 6890	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑m 𝑘))) → ((𝐺‘dom 𝑦)‘𝑦) = ((𝐺‘𝑘)‘𝑦))
9		fseqenlem.a	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
10		fseqenlem.b	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
11		fseqenlem.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:(𝐴 × 𝐴)–1-1-onto→𝐴)
12		fseqenlem.g	. . . . . . . . . . . 12 ⊢ 𝐺 = seqω((𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴 ↑m suc 𝑛) ↦ ((𝑓‘(𝑥 ↾ 𝑛))𝐹(𝑥‘𝑛)))), {⟨∅, 𝐵⟩})
13	9, 10, 11, 12	fseqenlem1 10027	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ω) → (𝐺‘𝑘):(𝐴 ↑m 𝑘)–1-1→𝐴)
14	13	adantrr 730	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑m 𝑘))) → (𝐺‘𝑘):(𝐴 ↑m 𝑘)–1-1→𝐴)
15		f1f 6781	. . . . . . . . . 10 ⊢ ((𝐺‘𝑘):(𝐴 ↑m 𝑘)–1-1→𝐴 → (𝐺‘𝑘):(𝐴 ↑m 𝑘)⟶𝐴)
16	14, 15	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑m 𝑘))) → (𝐺‘𝑘):(𝐴 ↑m 𝑘)⟶𝐴)
17		simprr 785	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑m 𝑘))) → 𝑦 ∈ (𝐴 ↑m 𝑘))
18	16, 17	ffvelcdmd 7087	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑m 𝑘))) → ((𝐺‘𝑘)‘𝑦) ∈ 𝐴)
19	8, 18	eqeltrd 2866	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑m 𝑘))) → ((𝐺‘dom 𝑦)‘𝑦) ∈ 𝐴)
20	6, 19	opelxpd 5705	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑m 𝑘))) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴))
21	20	rexlimdvaa 3170	. . . . 5 ⊢ (𝜑 → (∃𝑘 ∈ ω 𝑦 ∈ (𝐴 ↑m 𝑘) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴)))
22	1, 21	biimtrid 245	. . . 4 ⊢ (𝜑 → (𝑦 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴)))
23	22	imp 412	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘)) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴))
24		fseqenlem.k	. . 3 ⊢ 𝐾 = (𝑦 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘) ↦ ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩)
25	23, 24	fmptd 7116	. 2 ⊢ (𝜑 → 𝐾:∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘)⟶(ω × 𝐴))
26		ffun 6715	. . . . . . . . . . . . . . 15 ⊢ (𝐾:∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘)⟶(ω × 𝐴) → Fun 𝐾)
27		funbrfv2b 6945	. . . . . . . . . . . . . . 15 ⊢ (Fun 𝐾 → (𝑧𝐾𝑤 ↔ (𝑧 ∈ dom 𝐾 ∧ (𝐾‘𝑧) = 𝑤)))
28	25, 26, 27	3syl 19	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑧𝐾𝑤 ↔ (𝑧 ∈ dom 𝐾 ∧ (𝐾‘𝑧) = 𝑤)))
29	28	simplbda 505	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (𝐾‘𝑧) = 𝑤)
30	28	simprbda 504	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → 𝑧 ∈ dom 𝐾)
31	25	fdmd 6723	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom 𝐾 = ∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘))
32	31	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → dom 𝐾 = ∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘))
33	30, 32	eleqtrd 2868	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → 𝑧 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘))
34		dmeq 5898	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → dom 𝑦 = dom 𝑧)
35	34	fveq2d 6892	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → (𝐺‘dom 𝑦) = (𝐺‘dom 𝑧))
36		id 23	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧)
37	35, 36	fveq12d 6895	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → ((𝐺‘dom 𝑦)‘𝑦) = ((𝐺‘dom 𝑧)‘𝑧))
38	34, 37	opeq12d 4851	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑧 → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
39		opex 5450	. . . . . . . . . . . . . . 15 ⊢ ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩ ∈ V
40	38, 24, 39	fvmpt 6996	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘) → (𝐾‘𝑧) = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
41	33, 40	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (𝐾‘𝑧) = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
42	29, 41	eqtr3d 2803	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → 𝑤 = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
43	42	fveq2d 6892	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (1st ‘𝑤) = (1st ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩))
44		vex 3462	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
45	44	dmex 7915	. . . . . . . . . . . 12 ⊢ dom 𝑧 ∈ V
46		fvex 6901	. . . . . . . . . . . 12 ⊢ ((𝐺‘dom 𝑧)‘𝑧) ∈ V
47	45, 46	op1st 8003	. . . . . . . . . . 11 ⊢ (1st ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩) = dom 𝑧
48	43, 47	eqtrdi 2817	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (1st ‘𝑤) = dom 𝑧)
49	48	fveq2d 6892	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (𝐺‘(1st ‘𝑤)) = (𝐺‘dom 𝑧))
50	49	cnveqd 5866	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → ◡(𝐺‘(1st ‘𝑤)) = ◡(𝐺‘dom 𝑧))
51	42	fveq2d 6892	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (2nd ‘𝑤) = (2nd ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩))
52	45, 46	op2nd 8004	. . . . . . . . 9 ⊢ (2nd ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩) = ((𝐺‘dom 𝑧)‘𝑧)
53	51, 52	eqtrdi 2817	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (2nd ‘𝑤) = ((𝐺‘dom 𝑧)‘𝑧))
54	50, 53	fveq12d 6895	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (◡(𝐺‘(1st ‘𝑤))‘(2nd ‘𝑤)) = (◡(𝐺‘dom 𝑧)‘((𝐺‘dom 𝑧)‘𝑧)))
55		eliun 4965	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘) ↔ ∃𝑘 ∈ ω 𝑧 ∈ (𝐴 ↑m 𝑘))
56		elmapi 8855	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ (𝐴 ↑m 𝑘) → 𝑧:𝑘⟶𝐴)
57	56	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑m 𝑘)) → 𝑧:𝑘⟶𝐴)
58	57	fdmd 6723	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑m 𝑘)) → dom 𝑧 = 𝑘)
59		simpl 488	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑m 𝑘)) → 𝑘 ∈ ω)
60	58, 59	eqeltrd 2866	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑m 𝑘)) → dom 𝑧 ∈ ω)
61		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑m 𝑘)) → 𝑧 ∈ (𝐴 ↑m 𝑘))
62	58	oveq2d 7439	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑m 𝑘)) → (𝐴 ↑m dom 𝑧) = (𝐴 ↑m 𝑘))
63	61, 62	eleqtrrd 2869	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑m 𝑘)) → 𝑧 ∈ (𝐴 ↑m dom 𝑧))
64	60, 63	jca 521	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑m 𝑘)) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑m dom 𝑧)))
65	64	rexlimiva 3161	. . . . . . . . . . . . 13 ⊢ (∃𝑘 ∈ ω 𝑧 ∈ (𝐴 ↑m 𝑘) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑m dom 𝑧)))
66	55, 65	sylbi 220	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑m dom 𝑧)))
67	33, 66	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑m dom 𝑧)))
68	67	simpld 500	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → dom 𝑧 ∈ ω)
69	9, 10, 11, 12	fseqenlem1 10027	. . . . . . . . . 10 ⊢ ((𝜑 ∧ dom 𝑧 ∈ ω) → (𝐺‘dom 𝑧):(𝐴 ↑m dom 𝑧)–1-1→𝐴)
70	68, 69	syldan 603	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (𝐺‘dom 𝑧):(𝐴 ↑m dom 𝑧)–1-1→𝐴)
71		f1f1orn 6839	. . . . . . . . 9 ⊢ ((𝐺‘dom 𝑧):(𝐴 ↑m dom 𝑧)–1-1→𝐴 → (𝐺‘dom 𝑧):(𝐴 ↑m dom 𝑧)–1-1-onto→ran (𝐺‘dom 𝑧))
72	70, 71	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (𝐺‘dom 𝑧):(𝐴 ↑m dom 𝑧)–1-1-onto→ran (𝐺‘dom 𝑧))
73	67	simprd 501	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → 𝑧 ∈ (𝐴 ↑m dom 𝑧))
74		f1ocnvfv1 7285	. . . . . . . 8 ⊢ (((𝐺‘dom 𝑧):(𝐴 ↑m dom 𝑧)–1-1-onto→ran (𝐺‘dom 𝑧) ∧ 𝑧 ∈ (𝐴 ↑m dom 𝑧)) → (◡(𝐺‘dom 𝑧)‘((𝐺‘dom 𝑧)‘𝑧)) = 𝑧)
75	72, 73, 74	syl2anc 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (◡(𝐺‘dom 𝑧)‘((𝐺‘dom 𝑧)‘𝑧)) = 𝑧)
76	54, 75	eqtr2d 2802	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → 𝑧 = (◡(𝐺‘(1st ‘𝑤))‘(2nd ‘𝑤)))
77	76	ex 418	. . . . 5 ⊢ (𝜑 → (𝑧𝐾𝑤 → 𝑧 = (◡(𝐺‘(1st ‘𝑤))‘(2nd ‘𝑤))))
78	77	alrimiv 1960	. . . 4 ⊢ (𝜑 → ∀𝑧(𝑧𝐾𝑤 → 𝑧 = (◡(𝐺‘(1st ‘𝑤))‘(2nd ‘𝑤))))
79		mo2icl 3680	. . . 4 ⊢ (∀𝑧(𝑧𝐾𝑤 → 𝑧 = (◡(𝐺‘(1st ‘𝑤))‘(2nd ‘𝑤))) → ∃*𝑧 𝑧𝐾𝑤)
80	78, 79	syl 18	. . 3 ⊢ (𝜑 → ∃*𝑧 𝑧𝐾𝑤)
81	80	alrimiv 1960	. 2 ⊢ (𝜑 → ∀𝑤∃*𝑧 𝑧𝐾𝑤)
82		dff12 6780	. 2 ⊢ (𝐾:∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘)–1-1→(ω × 𝐴) ↔ (𝐾:∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘)⟶(ω × 𝐴) ∧ ∀𝑤∃*𝑧 𝑧𝐾𝑤))
83	25, 81, 82	sylanbrc 595	1 ⊢ (𝜑 → 𝐾:∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘)–1-1→(ω × 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2146  ∃*wmo 2568  ∃wrex 3092  Vcvv 3458  ∅c0 4289  {csn 4594  ⟨cop 4600  ∪ ciun 4961   class class class wbr 5114   ↦ cmpt 5197   × cxp 5664  ^◡ccnv 5665  dom cdm 5666  ran crn 5667   ↾ cres 5668  suc csuc 6369  Fun wfun 6537  ⟶wf 6539  –1-1→wf1 6540  –1-1-onto→wf1o 6542  ‘cfv 6543  (class class class)co 7423   ∈ cmpo 7425  ωcom 7871  1^st c1st 7993  2^nd c2nd 7994  seq_ωcseqom 8443   ↑_m cmap 8833

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-seqom 8444  df-1o 8462  df-map 8835

This theorem is used by:  fseqen  10030  pwfseqlem5  10666