Theorem fseqenlem2 8848

Description: Lemma for fseqen 8850. (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 4524	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘) ↔ ∃𝑘 ∈ ω 𝑦 ∈ (𝐴 ↑𝑚 𝑘))
2		elmapi 7879	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐴 ↑𝑚 𝑘) → 𝑦:𝑘⟶𝐴)
3	2	ad2antll 765	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑𝑚 𝑘))) → 𝑦:𝑘⟶𝐴)
4		fdm 6051	. . . . . . . . 9 ⊢ (𝑦:𝑘⟶𝐴 → dom 𝑦 = 𝑘)
5	3, 4	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑𝑚 𝑘))) → dom 𝑦 = 𝑘)
6		simprl 794	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑𝑚 𝑘))) → 𝑘 ∈ ω)
7	5, 6	eqeltrd 2701	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑𝑚 𝑘))) → dom 𝑦 ∈ ω)
8	5	fveq2d 6195	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑𝑚 𝑘))) → (𝐺‘dom 𝑦) = (𝐺‘𝑘))
9	8	fveq1d 6193	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑𝑚 𝑘))) → ((𝐺‘dom 𝑦)‘𝑦) = ((𝐺‘𝑘)‘𝑦))
10		fseqenlem.a	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
11		fseqenlem.b	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
12		fseqenlem.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:(𝐴 × 𝐴)–1-1-onto→𝐴)
13		fseqenlem.g	. . . . . . . . . . . 12 ⊢ 𝐺 = seq𝜔((𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴 ↑𝑚 suc 𝑛) ↦ ((𝑓‘(𝑥 ↾ 𝑛))𝐹(𝑥‘𝑛)))), {⟨∅, 𝐵⟩})
14	10, 11, 12, 13	fseqenlem1 8847	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ω) → (𝐺‘𝑘):(𝐴 ↑𝑚 𝑘)–1-1→𝐴)
15	14	adantrr 753	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑𝑚 𝑘))) → (𝐺‘𝑘):(𝐴 ↑𝑚 𝑘)–1-1→𝐴)
16		f1f 6101	. . . . . . . . . 10 ⊢ ((𝐺‘𝑘):(𝐴 ↑𝑚 𝑘)–1-1→𝐴 → (𝐺‘𝑘):(𝐴 ↑𝑚 𝑘)⟶𝐴)
17	15, 16	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑𝑚 𝑘))) → (𝐺‘𝑘):(𝐴 ↑𝑚 𝑘)⟶𝐴)
18		simprr 796	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑𝑚 𝑘))) → 𝑦 ∈ (𝐴 ↑𝑚 𝑘))
19	17, 18	ffvelrnd 6360	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑𝑚 𝑘))) → ((𝐺‘𝑘)‘𝑦) ∈ 𝐴)
20	9, 19	eqeltrd 2701	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑𝑚 𝑘))) → ((𝐺‘dom 𝑦)‘𝑦) ∈ 𝐴)
21		opelxpi 5148	. . . . . . 7 ⊢ ((dom 𝑦 ∈ ω ∧ ((𝐺‘dom 𝑦)‘𝑦) ∈ 𝐴) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴))
22	7, 20, 21	syl2anc 693	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴 ↑𝑚 𝑘))) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴))
23	22	rexlimdvaa 3032	. . . . 5 ⊢ (𝜑 → (∃𝑘 ∈ ω 𝑦 ∈ (𝐴 ↑𝑚 𝑘) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴)))
24	1, 23	syl5bi 232	. . . 4 ⊢ (𝜑 → (𝑦 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴)))
25	24	imp 445	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘)) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴))
26		fseqenlem.k	. . 3 ⊢ 𝐾 = (𝑦 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘) ↦ ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩)
27	25, 26	fmptd 6385	. 2 ⊢ (𝜑 → 𝐾:∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘)⟶(ω × 𝐴))
28		ffun 6048	. . . . . . . . . . . . . . 15 ⊢ (𝐾:∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘)⟶(ω × 𝐴) → Fun 𝐾)
29		funbrfv2b 6240	. . . . . . . . . . . . . . 15 ⊢ (Fun 𝐾 → (𝑧𝐾𝑤 ↔ (𝑧 ∈ dom 𝐾 ∧ (𝐾‘𝑧) = 𝑤)))
30	27, 28, 29	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑧𝐾𝑤 ↔ (𝑧 ∈ dom 𝐾 ∧ (𝐾‘𝑧) = 𝑤)))
31	30	simplbda 654	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (𝐾‘𝑧) = 𝑤)
32	30	simprbda 653	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → 𝑧 ∈ dom 𝐾)
33		fdm 6051	. . . . . . . . . . . . . . . . 17 ⊢ (𝐾:∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘)⟶(ω × 𝐴) → dom 𝐾 = ∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘))
34	27, 33	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom 𝐾 = ∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘))
35	34	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → dom 𝐾 = ∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘))
36	32, 35	eleqtrd 2703	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → 𝑧 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘))
37		dmeq 5324	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → dom 𝑦 = dom 𝑧)
38	37	fveq2d 6195	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → (𝐺‘dom 𝑦) = (𝐺‘dom 𝑧))
39		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧)
40	38, 39	fveq12d 6197	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → ((𝐺‘dom 𝑦)‘𝑦) = ((𝐺‘dom 𝑧)‘𝑧))
41	37, 40	opeq12d 4410	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑧 → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
42		opex 4932	. . . . . . . . . . . . . . 15 ⊢ ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩ ∈ V
43	41, 26, 42	fvmpt 6282	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘) → (𝐾‘𝑧) = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
44	36, 43	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (𝐾‘𝑧) = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
45	31, 44	eqtr3d 2658	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → 𝑤 = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
46	45	fveq2d 6195	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (1st ‘𝑤) = (1st ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩))
47		vex 3203	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
48	47	dmex 7099	. . . . . . . . . . . 12 ⊢ dom 𝑧 ∈ V
49		fvex 6201	. . . . . . . . . . . 12 ⊢ ((𝐺‘dom 𝑧)‘𝑧) ∈ V
50	48, 49	op1st 7176	. . . . . . . . . . 11 ⊢ (1st ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩) = dom 𝑧
51	46, 50	syl6eq 2672	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (1st ‘𝑤) = dom 𝑧)
52	51	fveq2d 6195	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (𝐺‘(1st ‘𝑤)) = (𝐺‘dom 𝑧))
53	52	cnveqd 5298	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → ◡(𝐺‘(1st ‘𝑤)) = ◡(𝐺‘dom 𝑧))
54	45	fveq2d 6195	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (2nd ‘𝑤) = (2nd ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩))
55	48, 49	op2nd 7177	. . . . . . . . 9 ⊢ (2nd ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩) = ((𝐺‘dom 𝑧)‘𝑧)
56	54, 55	syl6eq 2672	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (2nd ‘𝑤) = ((𝐺‘dom 𝑧)‘𝑧))
57	53, 56	fveq12d 6197	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (◡(𝐺‘(1st ‘𝑤))‘(2nd ‘𝑤)) = (◡(𝐺‘dom 𝑧)‘((𝐺‘dom 𝑧)‘𝑧)))
58		eliun 4524	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘) ↔ ∃𝑘 ∈ ω 𝑧 ∈ (𝐴 ↑𝑚 𝑘))
59		elmapi 7879	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ (𝐴 ↑𝑚 𝑘) → 𝑧:𝑘⟶𝐴)
60	59	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑𝑚 𝑘)) → 𝑧:𝑘⟶𝐴)
61		fdm 6051	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧:𝑘⟶𝐴 → dom 𝑧 = 𝑘)
62	60, 61	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑𝑚 𝑘)) → dom 𝑧 = 𝑘)
63		simpl 473	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑𝑚 𝑘)) → 𝑘 ∈ ω)
64	62, 63	eqeltrd 2701	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑𝑚 𝑘)) → dom 𝑧 ∈ ω)
65		simpr 477	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑𝑚 𝑘)) → 𝑧 ∈ (𝐴 ↑𝑚 𝑘))
66	62	oveq2d 6666	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑𝑚 𝑘)) → (𝐴 ↑𝑚 dom 𝑧) = (𝐴 ↑𝑚 𝑘))
67	65, 66	eleqtrrd 2704	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑𝑚 𝑘)) → 𝑧 ∈ (𝐴 ↑𝑚 dom 𝑧))
68	64, 67	jca 554	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑𝑚 𝑘)) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑𝑚 dom 𝑧)))
69	68	rexlimiva 3028	. . . . . . . . . . . . 13 ⊢ (∃𝑘 ∈ ω 𝑧 ∈ (𝐴 ↑𝑚 𝑘) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑𝑚 dom 𝑧)))
70	58, 69	sylbi 207	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑𝑚 dom 𝑧)))
71	36, 70	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴 ↑𝑚 dom 𝑧)))
72	71	simpld 475	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → dom 𝑧 ∈ ω)
73	10, 11, 12, 13	fseqenlem1 8847	. . . . . . . . . 10 ⊢ ((𝜑 ∧ dom 𝑧 ∈ ω) → (𝐺‘dom 𝑧):(𝐴 ↑𝑚 dom 𝑧)–1-1→𝐴)
74	72, 73	syldan 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (𝐺‘dom 𝑧):(𝐴 ↑𝑚 dom 𝑧)–1-1→𝐴)
75		f1f1orn 6148	. . . . . . . . 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))
76	74, 75	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (𝐺‘dom 𝑧):(𝐴 ↑𝑚 dom 𝑧)–1-1-onto→ran (𝐺‘dom 𝑧))
77	71	simprd 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → 𝑧 ∈ (𝐴 ↑𝑚 dom 𝑧))
78		f1ocnvfv1 6532	. . . . . . . 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)
79	76, 77, 78	syl2anc 693	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → (◡(𝐺‘dom 𝑧)‘((𝐺‘dom 𝑧)‘𝑧)) = 𝑧)
80	57, 79	eqtr2d 2657	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧𝐾𝑤) → 𝑧 = (◡(𝐺‘(1st ‘𝑤))‘(2nd ‘𝑤)))
81	80	ex 450	. . . . 5 ⊢ (𝜑 → (𝑧𝐾𝑤 → 𝑧 = (◡(𝐺‘(1st ‘𝑤))‘(2nd ‘𝑤))))
82	81	alrimiv 1855	. . . 4 ⊢ (𝜑 → ∀𝑧(𝑧𝐾𝑤 → 𝑧 = (◡(𝐺‘(1st ‘𝑤))‘(2nd ‘𝑤))))
83		mo2icl 3385	. . . 4 ⊢ (∀𝑧(𝑧𝐾𝑤 → 𝑧 = (◡(𝐺‘(1st ‘𝑤))‘(2nd ‘𝑤))) → ∃*𝑧 𝑧𝐾𝑤)
84	82, 83	syl 17	. . 3 ⊢ (𝜑 → ∃*𝑧 𝑧𝐾𝑤)
85	84	alrimiv 1855	. 2 ⊢ (𝜑 → ∀𝑤∃*𝑧 𝑧𝐾𝑤)
86		dff12 6100	. 2 ⊢ (𝐾:∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘)–1-1→(ω × 𝐴) ↔ (𝐾:∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘)⟶(ω × 𝐴) ∧ ∀𝑤∃*𝑧 𝑧𝐾𝑤))
87	27, 85, 86	sylanbrc 698	1 ⊢ (𝜑 → 𝐾:∪ 𝑘 ∈ ω (𝐴 ↑𝑚 𝑘)–1-1→(ω × 𝐴))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481   = wceq 1483   ∈ wcel 1990  ∃*wmo 2471  ∃wrex 2913  Vcvv 3200  ∅c0 3915  {csn 4177  ⟨cop 4183  ∪ ciun 4520   class class class wbr 4653   ↦ cmpt 4729   × cxp 5112  ^◡ccnv 5113  dom cdm 5114  ran crn 5115   ↾ cres 5116  suc csuc 5725  Fun wfun 5882  ⟶wf 5884  –1-1→wf1 5885  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  ωcom 7065  1^st c1st 7166  2^nd c2nd 7167  seq_𝜔cseqom 7542   ↑_𝑚 cmap 7857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-1o 7560  df-map 7859

This theorem is referenced by:  fseqen  8850  pwfseqlem5  9485