Theorem fseqenlem2 9445

Description: Lemma for fseqen 9447. (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 4915	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘) ↔ ∃𝑘 ∈ ω 𝑦 ∈ (𝐴 ↑m 𝑘))
2		elmapi 8422	. . . . . . . . . 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 6517	. . . . . . . 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 2913	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑m 𝑘))) → dom 𝑦 ∈ ω)
7	4	fveq2d 6668	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑m 𝑘))) → (𝐺‘dom 𝑦) = (𝐺‘𝑘))
8	7	fveq1d 6666	. . . . . . . 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 9444	. . . . . . . . . . 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 6569	. . . . . . . . . 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	ffvelrnd 6846	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑m 𝑘))) → ((𝐺‘𝑘)‘𝑦) ∈ 𝐴)
19	8, 18	eqeltrd 2913	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑m 𝑘))) → ((𝐺‘dom 𝑦)‘𝑦) ∈ 𝐴)
20	6, 19	opelxpd 5587	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑m 𝑘))) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴))
21	20	rexlimdvaa 3285	. . . . 5 ⊢ (𝜑 → (∃𝑘 ∈ ω 𝑦 ∈ (𝐴 ↑m 𝑘) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴)))
22	1, 21	syl5bi 244	. . . 4 ⊢ (𝜑 → (𝑦 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴)))
23	22	imp 409	. . 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 6872	. 2 ⊢ (𝜑 → 𝐾:∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘)⟶(ω × 𝐴))
26		ffun 6511	. . . . . . . . . . . . . . 15 ⊢ (𝐾:∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘)⟶(ω × 𝐴) → Fun 𝐾)
27		funbrfv2b 6717	. . . . . . . . . . . . . . 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 502	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (𝐾‘𝑧) = 𝑤)
30	28	simprbda 501	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → 𝑧 ∈ dom 𝐾)
31	25	fdmd 6517	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom 𝐾 = ∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘))
32	31	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → dom 𝐾 = ∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘))
33	30, 32	eleqtrd 2915	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → 𝑧 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘))
34		dmeq 5766	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → dom 𝑦 = dom 𝑧)
35	34	fveq2d 6668	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → (𝐺‘dom 𝑦) = (𝐺‘dom 𝑧))
36		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧)
37	35, 36	fveq12d 6671	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → ((𝐺‘dom 𝑦)‘𝑦) = ((𝐺‘dom 𝑧)‘𝑧))
38	34, 37	opeq12d 4804	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑧 → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
39		opex 5348	. . . . . . . . . . . . . . 15 ⊢ ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩ ∈ V
40	38, 24, 39	fvmpt 6762	. . . . . . . . . . . . . 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 2858	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → 𝑤 = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
43	42	fveq2d 6668	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (1st ‘𝑤) = (1st ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩))
44		vex 3497	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
45	44	dmex 7610	. . . . . . . . . . . 12 ⊢ dom 𝑧 ∈ V
46		fvex 6677	. . . . . . . . . . . 12 ⊢ ((𝐺‘dom 𝑧)‘𝑧) ∈ V
47	45, 46	op1st 7691	. . . . . . . . . . 11 ⊢ (1st ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩) = dom 𝑧
48	43, 47	syl6eq 2872	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (1st ‘𝑤) = dom 𝑧)
49	48	fveq2d 6668	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (𝐺‘(1st ‘𝑤)) = (𝐺‘dom 𝑧))
50	49	cnveqd 5740	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → ◡(𝐺‘(1st ‘𝑤)) = ◡(𝐺‘dom 𝑧))
51	42	fveq2d 6668	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (2nd ‘𝑤) = (2nd ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩))
52	45, 46	op2nd 7692	. . . . . . . . 9 ⊢ (2nd ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩) = ((𝐺‘dom 𝑧)‘𝑧)
53	51, 52	syl6eq 2872	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (2nd ‘𝑤) = ((𝐺‘dom 𝑧)‘𝑧))
54	50, 53	fveq12d 6671	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (◡(𝐺‘(1st ‘𝑤))‘(2nd ‘𝑤)) = (◡(𝐺‘dom 𝑧)‘((𝐺‘dom 𝑧)‘𝑧)))
55		eliun 4915	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘) ↔ ∃𝑘 ∈ ω 𝑧 ∈ (𝐴 ↑m 𝑘))
56		elmapi 8422	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ (𝐴 ↑m 𝑘) → 𝑧:𝑘⟶𝐴)
57	56	adantl 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑m 𝑘)) → 𝑧:𝑘⟶𝐴)
58	57	fdmd 6517	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑m 𝑘)) → dom 𝑧 = 𝑘)
59		simpl 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑m 𝑘)) → 𝑘 ∈ ω)
60	58, 59	eqeltrd 2913	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑m 𝑘)) → dom 𝑧 ∈ ω)
61		simpr 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑m 𝑘)) → 𝑧 ∈ (𝐴 ↑m 𝑘))
62	58	oveq2d 7166	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑m 𝑘)) → (𝐴 ↑m dom 𝑧) = (𝐴 ↑m 𝑘))
63	61, 62	eleqtrrd 2916	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑m 𝑘)) → 𝑧 ∈ (𝐴 ↑m dom 𝑧))
64	60, 63	jca 514	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑m 𝑘)) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑m dom 𝑧)))
65	64	rexlimiva 3281	. . . . . . . . . . . . 13 ⊢ (∃𝑘 ∈ ω 𝑧 ∈ (𝐴 ↑m 𝑘) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑m dom 𝑧)))
66	55, 65	sylbi 219	. . . . . . . . . . . 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 497	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → dom 𝑧 ∈ ω)
69	9, 10, 11, 12	fseqenlem1 9444	. . . . . . . . . 10 ⊢ ((𝜑 ∧ dom 𝑧 ∈ ω) → (𝐺‘dom 𝑧):(𝐴 ↑m dom 𝑧)–1-1→𝐴)
70	68, 69	syldan 593	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (𝐺‘dom 𝑧):(𝐴 ↑m dom 𝑧)–1-1→𝐴)
71		f1f1orn 6620	. . . . . . . . 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 498	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → 𝑧 ∈ (𝐴 ↑m dom 𝑧))
74		f1ocnvfv1 7027	. . . . . . . 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 586	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (◡(𝐺‘dom 𝑧)‘((𝐺‘dom 𝑧)‘𝑧)) = 𝑧)
76	54, 75	eqtr2d 2857	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → 𝑧 = (◡(𝐺‘(1st ‘𝑤))‘(2nd ‘𝑤)))
77	76	ex 415	. . . . 5 ⊢ (𝜑 → (𝑧𝐾𝑤 → 𝑧 = (◡(𝐺‘(1st ‘𝑤))‘(2nd ‘𝑤))))
78	77	alrimiv 1924	. . . 4 ⊢ (𝜑 → ∀𝑧(𝑧𝐾𝑤 → 𝑧 = (◡(𝐺‘(1st ‘𝑤))‘(2nd ‘𝑤))))
79		mo2icl 3704	. . . 4 ⊢ (∀𝑧(𝑧𝐾𝑤 → 𝑧 = (◡(𝐺‘(1st ‘𝑤))‘(2nd ‘𝑤))) → ∃*𝑧 𝑧𝐾𝑤)
80	78, 79	syl 17	. . 3 ⊢ (𝜑 → ∃*𝑧 𝑧𝐾𝑤)
81	80	alrimiv 1924	. 2 ⊢ (𝜑 → ∀𝑤∃*𝑧 𝑧𝐾𝑤)
82		dff12 6568	. 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 585	1 ⊢ (𝜑 → 𝐾:∪ 𝑘 ∈ ω (𝐴 ↑m 𝑘)–1-1→(ω × 𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1531   = wceq 1533   ∈ wcel 2110  ∃*wmo 2616  ∃wrex 3139  Vcvv 3494  ∅c0 4290  {csn 4560  ⟨cop 4566  ∪ ciun 4911   class class class wbr 5058   ↦ cmpt 5138   × cxp 5547  ^◡ccnv 5548  dom cdm 5549  ran crn 5550   ↾ cres 5551  suc csuc 6187  Fun wfun 6343  ⟶wf 6345  –1-1→wf1 6346  –1-1-onto→wf1o 6348  ‘cfv 6349  (class class class)co 7150   ∈ cmpo 7152  ωcom 7574  1^st c1st 7681  2^nd c2nd 7682  seq_ωcseqom 8077   ↑_m cmap 8400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-seqom 8078  df-1o 8096  df-map 8402

This theorem is referenced by:  fseqen  9447  pwfseqlem5  10079