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Theorem fseqenlem2 10134

Description: Lemma for fseqen 10136. (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)⟶1-1onto𝐴)
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 4958	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑_m 𝑘) ↔ ∃𝑘 ∈ ω 𝑦 ∈ (𝐴 ↑_m 𝑘))
2		elmapi 8900	. . . . . . . . . 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 6745	. . . . . . . 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 2864	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑_m 𝑘))) → dom 𝑦 ∈ ω)
7	4	fveq2d 6915	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑_m 𝑘))) → (𝐺‘dom 𝑦) = (𝐺‘𝑘))
8	7	fveq1d 6913	. . . . . . . 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)⟶1-1onto𝐴)
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 10133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ω) → (𝐺‘𝑘):(𝐴 ↑_m 𝑘)⟶1-1𝐴)
14	13	adantrr 730	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑_m 𝑘))) → (𝐺‘𝑘):(𝐴 ↑_m 𝑘)⟶1-1𝐴)
15		f1f 6803	. . . . . . . . . 10 ⊢ ((𝐺‘𝑘):(𝐴 ↑_m 𝑘)⟶1-1𝐴 → (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 7112	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑_m 𝑘))) → ((𝐺‘𝑘)‘𝑦) ∈ 𝐴)
19	8, 18	eqeltrd 2864	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑_m 𝑘))) → ((𝐺‘dom 𝑦)‘𝑦) ∈ 𝐴)
20	6, 19	opelxpd 5695	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑_m 𝑘))) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴))
21	20	rexlimdvaa 3168	. . . . 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 7141	. 2 ⊢ (𝜑 → 𝐾:∪ 𝑘 ∈ ω (𝐴 ↑_m 𝑘)⟶(ω × 𝐴))
26		ffun 6737	. . . . . . . . . . . . . . 15 ⊢ (𝐾:∪ 𝑘 ∈ ω (𝐴 ↑_m 𝑘)⟶(ω × 𝐴) → Fun 𝐾)
27		funbrfv2b 6968	. . . . . . . . . . . . . . 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 6745	. . . . . . . . . . . . . . . 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 2866	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → 𝑧 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑_m 𝑘))
34		dmeq 5893	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → dom 𝑦 = dom 𝑧)
35	34	fveq2d 6915	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → (𝐺‘dom 𝑦) = (𝐺‘dom 𝑧))
36		id 23	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧)
37	35, 36	fveq12d 6918	. . . . . . . . . . . . . . . 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 5437	. . . . . . . . . . . . . . 15 ⊢ ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩ ∈ V
40	38, 24, 39	fvmpt 7019	. . . . . . . . . . . . . 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 2800	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → 𝑤 = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
43	42	fveq2d 6915	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (1^st ‘𝑤) = (1^st ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩))
44		vex 3458	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
45	44	dmex 7948	. . . . . . . . . . . 12 ⊢ dom 𝑧 ∈ V
46		fvex 6924	. . . . . . . . . . . 12 ⊢ ((𝐺‘dom 𝑧)‘𝑧) ∈ V
47	45, 46	op1st 8036	. . . . . . . . . . 11 ⊢ (1^st ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩) = dom 𝑧
48	43, 47	eqtrdi 2814	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (1^st ‘𝑤) = dom 𝑧)
49	48	fveq2d 6915	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (𝐺‘(1^st ‘𝑤)) = (𝐺‘dom 𝑧))
50	49	cnveqd 5861	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → ^◡(𝐺‘(1^st ‘𝑤)) = ^◡(𝐺‘dom 𝑧))
51	42	fveq2d 6915	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (2^nd ‘𝑤) = (2^nd ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩))
52	45, 46	op2nd 8037	. . . . . . . . 9 ⊢ (2^nd ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩) = ((𝐺‘dom 𝑧)‘𝑧)
53	51, 52	eqtrdi 2814	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (2^nd ‘𝑤) = ((𝐺‘dom 𝑧)‘𝑧))
54	50, 53	fveq12d 6918	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (^◡(𝐺‘(1^st ‘𝑤))‘(2^nd ‘𝑤)) = (^◡(𝐺‘dom 𝑧)‘((𝐺‘dom 𝑧)‘𝑧)))
55		eliun 4958	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑_m 𝑘) ↔ ∃𝑘 ∈ ω 𝑧 ∈ (𝐴 ↑_m 𝑘))
56		elmapi 8900	. . . . . . . . . . . . . . . . . 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 6745	. . . . . . . . . . . . . . . 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 2864	. . . . . . . . . . . . . . 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 7462	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑_m 𝑘)) → (𝐴 ↑_m dom 𝑧) = (𝐴 ↑_m 𝑘))
63	61, 62	eleqtrrd 2867	. . . . . . . . . . . . . . 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 3159	. . . . . . . . . . . . 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 10133	. . . . . . . . . 10 ⊢ ((𝜑 ∧ dom 𝑧 ∈ ω) → (𝐺‘dom 𝑧):(𝐴 ↑_m dom 𝑧)⟶1-1𝐴)
70	68, 69	syldan 603	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (𝐺‘dom 𝑧):(𝐴 ↑_m dom 𝑧)⟶1-1𝐴)
71		f1f1orn 6861	. . . . . . . . 9 ⊢ ((𝐺‘dom 𝑧):(𝐴 ↑_m dom 𝑧)⟶1-1𝐴 → (𝐺‘dom 𝑧):(𝐴 ↑_m dom 𝑧)⟶1-1ontoran (𝐺‘dom 𝑧))
72	70, 71	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (𝐺‘dom 𝑧):(𝐴 ↑_m dom 𝑧)⟶1-1ontoran (𝐺‘dom 𝑧))
73	67	simprd 501	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → 𝑧 ∈ (𝐴 ↑_m dom 𝑧))
74		f1ocnvfv1 7310	. . . . . . . 8 ⊢ (((𝐺‘dom 𝑧):(𝐴 ↑_m dom 𝑧)⟶1-1ontoran (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 2799	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → 𝑧 = (^◡(𝐺‘(1^st ‘𝑤))‘(2^nd ‘𝑤)))
77	76	ex 418	. . . . 5 ⊢ (𝜑 → (𝑧𝐾𝑤 → 𝑧 = (^◡(𝐺‘(1^st ‘𝑤))‘(2^nd ‘𝑤))))
78	77	alrimiv 1962	. . . 4 ⊢ (𝜑 → ∀𝑧(𝑧𝐾𝑤 → 𝑧 = (^◡(𝐺‘(1^st ‘𝑤))‘(2^nd ‘𝑤))))
79		mo2icl 3675	. . . 4 ⊢ (∀𝑧(𝑧𝐾𝑤 → 𝑧 = (^◡(𝐺‘(1^st ‘𝑤))‘(2^nd ‘𝑤))) → ∃*𝑧 𝑧𝐾𝑤)
80	78, 79	syl 18	. . 3 ⊢ (𝜑 → ∃*𝑧 𝑧𝐾𝑤)
81	80	alrimiv 1962	. 2 ⊢ (𝜑 → ∀𝑤∃*𝑧 𝑧𝐾𝑤)
82		dff12 6802	. 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))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∀wal 1568 = wceq 1570 ∈ wcel 2147 ∃*wmo 2565 ∃wrex 3090 Vcvv 3454 ∅c0 4281 {csn 4588 ⟨cop 4838 ∪ ciun 4954 class class class wbr 5106 ↦ cmpt 5190 × cxp 5654 ^◡ccnv 5655 dom cdm 5656 ran crn 5657 ↾ cres 5658 suc csuc 6385 Fun wfun 6555 ⟶wf 6557 ⟶1-1wf1 6558 ⟶1-1ontowf1o 6560 ‘cfv 6561 (class class class)co 7446 ∈ cmpo 7448 ωcom 7904 1^st c1st 8026 2^nd c2nd 8027 seq_ωcseqom 8488 ↑_m cmap 8878

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5236 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7778

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6325 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7449 df-oprab 7450 df-mpo 7451 df-om 7905 df-1st 8028 df-2nd 8029 df-frecs 8330 df-wrecs 8361 df-recs 8410 df-rdg 8449 df-seqom 8489 df-1o 8507 df-map 8880

This theorem is used by: fseqen 10136 pwfseqlem5 10778

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