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Theorem fseqenlem2 10019

Description: Lemma for fseqen 10021. (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 = seq_ω((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 5001	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑_m 𝑘) ↔ ∃𝑘 ∈ ω 𝑦 ∈ (𝐴 ↑_m 𝑘))
2		elmapi 8842	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐴 ↑_m 𝑘) → 𝑦:𝑘⟶𝐴)
3	2	ad2antll 727	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑_m 𝑘))) → 𝑦:𝑘⟶𝐴)
4	3	fdmd 6728	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑_m 𝑘))) → dom 𝑦 = 𝑘)
5		simprl 769	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑_m 𝑘))) → 𝑘 ∈ ω)
6	4, 5	eqeltrd 2833	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑_m 𝑘))) → dom 𝑦 ∈ ω)
7	4	fveq2d 6895	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑_m 𝑘))) → (𝐺‘dom 𝑦) = (𝐺‘𝑘))
8	7	fveq1d 6893	. . . . . . . 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 = seq_ω((𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴 ↑_m suc 𝑛) ↦ ((𝑓‘(𝑥 ↾ 𝑛))𝐹(𝑥‘𝑛)))), {⟨∅, 𝐵⟩})
13	9, 10, 11, 12	fseqenlem1 10018	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ω) → (𝐺‘𝑘):(𝐴 ↑_m 𝑘)–1-1→𝐴)
14	13	adantrr 715	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑_m 𝑘))) → (𝐺‘𝑘):(𝐴 ↑_m 𝑘)–1-1→𝐴)
15		f1f 6787	. . . . . . . . . 10 ⊢ ((𝐺‘𝑘):(𝐴 ↑_m 𝑘)–1-1→𝐴 → (𝐺‘𝑘):(𝐴 ↑_m 𝑘)⟶𝐴)
16	14, 15	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑_m 𝑘))) → (𝐺‘𝑘):(𝐴 ↑_m 𝑘)⟶𝐴)
17		simprr 771	. . . . . . . . 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 2833	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑_m 𝑘))) → ((𝐺‘dom 𝑦)‘𝑦) ∈ 𝐴)
20	6, 19	opelxpd 5715	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑_m 𝑘))) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴))
21	20	rexlimdvaa 3156	. . . . 5 ⊢ (𝜑 → (∃𝑘 ∈ ω 𝑦 ∈ (𝐴 ↑_m 𝑘) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴)))
22	1, 21	biimtrid 241	. . . 4 ⊢ (𝜑 → (𝑦 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑_m 𝑘) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴)))
23	22	imp 407	. . 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 7113	. 2 ⊢ (𝜑 → 𝐾:∪ 𝑘 ∈ ω (𝐴 ↑_m 𝑘)⟶(ω × 𝐴))
26		ffun 6720	. . . . . . . . . . . . . . 15 ⊢ (𝐾:∪ 𝑘 ∈ ω (𝐴 ↑_m 𝑘)⟶(ω × 𝐴) → Fun 𝐾)
27		funbrfv2b 6949	. . . . . . . . . . . . . . 15 ⊢ (Fun 𝐾 → (𝑧𝐾𝑤 ↔ (𝑧 ∈ dom 𝐾 ∧ (𝐾‘𝑧) = 𝑤)))
28	25, 26, 27	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑧𝐾𝑤 ↔ (𝑧 ∈ dom 𝐾 ∧ (𝐾‘𝑧) = 𝑤)))
29	28	simplbda 500	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (𝐾‘𝑧) = 𝑤)
30	28	simprbda 499	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → 𝑧 ∈ dom 𝐾)
31	25	fdmd 6728	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom 𝐾 = ∪ 𝑘 ∈ ω (𝐴 ↑_m 𝑘))
32	31	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → dom 𝐾 = ∪ 𝑘 ∈ ω (𝐴 ↑_m 𝑘))
33	30, 32	eleqtrd 2835	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → 𝑧 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑_m 𝑘))
34		dmeq 5903	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → dom 𝑦 = dom 𝑧)
35	34	fveq2d 6895	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → (𝐺‘dom 𝑦) = (𝐺‘dom 𝑧))
36		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧)
37	35, 36	fveq12d 6898	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → ((𝐺‘dom 𝑦)‘𝑦) = ((𝐺‘dom 𝑧)‘𝑧))
38	34, 37	opeq12d 4881	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑧 → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
39		opex 5464	. . . . . . . . . . . . . . 15 ⊢ ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩ ∈ V
40	38, 24, 39	fvmpt 6998	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑_m 𝑘) → (𝐾‘𝑧) = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
41	33, 40	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (𝐾‘𝑧) = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
42	29, 41	eqtr3d 2774	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → 𝑤 = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
43	42	fveq2d 6895	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (1^st ‘𝑤) = (1^st ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩))
44		vex 3478	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
45	44	dmex 7901	. . . . . . . . . . . 12 ⊢ dom 𝑧 ∈ V
46		fvex 6904	. . . . . . . . . . . 12 ⊢ ((𝐺‘dom 𝑧)‘𝑧) ∈ V
47	45, 46	op1st 7982	. . . . . . . . . . 11 ⊢ (1^st ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩) = dom 𝑧
48	43, 47	eqtrdi 2788	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (1^st ‘𝑤) = dom 𝑧)
49	48	fveq2d 6895	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (𝐺‘(1^st ‘𝑤)) = (𝐺‘dom 𝑧))
50	49	cnveqd 5875	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → ^◡(𝐺‘(1^st ‘𝑤)) = ^◡(𝐺‘dom 𝑧))
51	42	fveq2d 6895	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (2^nd ‘𝑤) = (2^nd ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩))
52	45, 46	op2nd 7983	. . . . . . . . 9 ⊢ (2^nd ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩) = ((𝐺‘dom 𝑧)‘𝑧)
53	51, 52	eqtrdi 2788	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (2^nd ‘𝑤) = ((𝐺‘dom 𝑧)‘𝑧))
54	50, 53	fveq12d 6898	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (^◡(𝐺‘(1^st ‘𝑤))‘(2^nd ‘𝑤)) = (^◡(𝐺‘dom 𝑧)‘((𝐺‘dom 𝑧)‘𝑧)))
55		eliun 5001	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑_m 𝑘) ↔ ∃𝑘 ∈ ω 𝑧 ∈ (𝐴 ↑_m 𝑘))
56		elmapi 8842	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ (𝐴 ↑_m 𝑘) → 𝑧:𝑘⟶𝐴)
57	56	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑_m 𝑘)) → 𝑧:𝑘⟶𝐴)
58	57	fdmd 6728	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑_m 𝑘)) → dom 𝑧 = 𝑘)
59		simpl 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑_m 𝑘)) → 𝑘 ∈ ω)
60	58, 59	eqeltrd 2833	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑_m 𝑘)) → dom 𝑧 ∈ ω)
61		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑_m 𝑘)) → 𝑧 ∈ (𝐴 ↑_m 𝑘))
62	58	oveq2d 7424	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑_m 𝑘)) → (𝐴 ↑_m dom 𝑧) = (𝐴 ↑_m 𝑘))
63	61, 62	eleqtrrd 2836	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑_m 𝑘)) → 𝑧 ∈ (𝐴 ↑_m dom 𝑧))
64	60, 63	jca 512	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑_m 𝑘)) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑_m dom 𝑧)))
65	64	rexlimiva 3147	. . . . . . . . . . . . 13 ⊢ (∃𝑘 ∈ ω 𝑧 ∈ (𝐴 ↑_m 𝑘) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑_m dom 𝑧)))
66	55, 65	sylbi 216	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑_m 𝑘) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑_m dom 𝑧)))
67	33, 66	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑_m dom 𝑧)))
68	67	simpld 495	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → dom 𝑧 ∈ ω)
69	9, 10, 11, 12	fseqenlem1 10018	. . . . . . . . . 10 ⊢ ((𝜑 ∧ dom 𝑧 ∈ ω) → (𝐺‘dom 𝑧):(𝐴 ↑_m dom 𝑧)–1-1→𝐴)
70	68, 69	syldan 591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (𝐺‘dom 𝑧):(𝐴 ↑_m dom 𝑧)–1-1→𝐴)
71		f1f1orn 6844	. . . . . . . . 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 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (𝐺‘dom 𝑧):(𝐴 ↑_m dom 𝑧)–1-1-onto→ran (𝐺‘dom 𝑧))
73	67	simprd 496	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → 𝑧 ∈ (𝐴 ↑_m dom 𝑧))
74		f1ocnvfv1 7273	. . . . . . . 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 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (^◡(𝐺‘dom 𝑧)‘((𝐺‘dom 𝑧)‘𝑧)) = 𝑧)
76	54, 75	eqtr2d 2773	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → 𝑧 = (^◡(𝐺‘(1^st ‘𝑤))‘(2^nd ‘𝑤)))
77	76	ex 413	. . . . 5 ⊢ (𝜑 → (𝑧𝐾𝑤 → 𝑧 = (^◡(𝐺‘(1^st ‘𝑤))‘(2^nd ‘𝑤))))
78	77	alrimiv 1930	. . . 4 ⊢ (𝜑 → ∀𝑧(𝑧𝐾𝑤 → 𝑧 = (^◡(𝐺‘(1^st ‘𝑤))‘(2^nd ‘𝑤))))
79		mo2icl 3710	. . . 4 ⊢ (∀𝑧(𝑧𝐾𝑤 → 𝑧 = (^◡(𝐺‘(1^st ‘𝑤))‘(2^nd ‘𝑤))) → ∃*𝑧 𝑧𝐾𝑤)
80	78, 79	syl 17	. . 3 ⊢ (𝜑 → ∃*𝑧 𝑧𝐾𝑤)
81	80	alrimiv 1930	. 2 ⊢ (𝜑 → ∀𝑤∃*𝑧 𝑧𝐾𝑤)
82		dff12 6786	. 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 583	1 ⊢ (𝜑 → 𝐾:∪ 𝑘 ∈ ω (𝐴 ↑_m 𝑘)–1-1→(ω × 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1539 = wceq 1541 ∈ wcel 2106 ∃*wmo 2532 ∃wrex 3070 Vcvv 3474 ∅c0 4322 {csn 4628 ⟨cop 4634 ∪ ciun 4997 class class class wbr 5148 ↦ cmpt 5231 × cxp 5674 ^◡ccnv 5675 dom cdm 5676 ran crn 5677 ↾ cres 5678 suc csuc 6366 Fun wfun 6537 ⟶wf 6539 –1-1→wf1 6540 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7408 ∈ cmpo 7410 ωcom 7854 1^st c1st 7972 2^nd c2nd 7973 seq_ωcseqom 8446 ↑_m cmap 8819

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-seqom 8447 df-1o 8465 df-map 8821

This theorem is referenced by: fseqen 10021 pwfseqlem5 10657

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