Theorem tfr1onlemres 6610

Description: Lemma for tfr1on 6611. Recursion is defined on an ordinal if the characteristic function is defined up to a suitable point. (Contributed by Jim Kingdon, 18-Mar-2022.)

Hypotheses

Ref	Expression
tfr1on.f	⊢ 𝐹 = recs(𝐺)
tfr1on.g	⊢ (𝜑 → Fun 𝐺)
tfr1on.x	⊢ (𝜑 → Ord 𝑋)
tfr1on.ex	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓 Fn 𝑥) → (𝐺‘𝑓) ∈ V)
tfr1onlemsucfn.1	⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
tfr1onlemres.u	⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
tfr1onlemres.yx	⊢ (𝜑 → 𝑌 ∈ 𝑋)

Assertion

Ref	Expression
tfr1onlemres	⊢ (𝜑 → 𝑌 ⊆ dom 𝐹)

Distinct variable groups:   𝑥,𝐴   𝑓,𝐺,𝑥,𝑦   𝑓,𝑋,𝑥   𝑓,𝑌,𝑥   𝜑,𝑓,𝑥

Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦,𝑓)   𝐹(𝑥,𝑦,𝑓)   𝑋(𝑦)   𝑌(𝑦)

Proof of Theorem tfr1onlemres

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

Step	Hyp	Ref	Expression
1		tfr1on.x	. . . . . . . . . 10 ⊢ (𝜑 → Ord 𝑋)
2	1	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑌) → Ord 𝑋)
3		simpr 110	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑌) → 𝑧 ∈ 𝑌)
4		tfr1onlemres.yx	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ 𝑋)
5	4	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑌) → 𝑌 ∈ 𝑋)
6	3, 5	jca 306	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑌) → (𝑧 ∈ 𝑌 ∧ 𝑌 ∈ 𝑋))
7		ordtr1 4528	. . . . . . . . 9 ⊢ (Ord 𝑋 → ((𝑧 ∈ 𝑌 ∧ 𝑌 ∈ 𝑋) → 𝑧 ∈ 𝑋))
8	2, 6, 7	sylc 62	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑌) → 𝑧 ∈ 𝑋)
9		tfr1on.f	. . . . . . . . 9 ⊢ 𝐹 = recs(𝐺)
10		tfr1on.g	. . . . . . . . 9 ⊢ (𝜑 → Fun 𝐺)
11		tfr1on.ex	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓 Fn 𝑥) → (𝐺‘𝑓) ∈ V)
12		tfr1onlemsucfn.1	. . . . . . . . 9 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
13		tfr1onlemres.u	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
14	9, 10, 1, 11, 12, 13	tfr1onlemaccex 6609	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑋) → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))
15	8, 14	syldan 282	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑌) → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))
16	10	ad2antrr 492	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → Fun 𝐺)
17	1	ad2antrr 492	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → Ord 𝑋)
18	11	3adant1r 1262	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋 ∧ 𝑓 Fn 𝑥) → (𝐺‘𝑓) ∈ V)
19	18	3adant1r 1262	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) ∧ 𝑥 ∈ 𝑋 ∧ 𝑓 Fn 𝑥) → (𝐺‘𝑓) ∈ V)
20	4	ad2antrr 492	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → 𝑌 ∈ 𝑋)
21	3	adantr 276	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → 𝑧 ∈ 𝑌)
22	13	adantlr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
23	22	adantlr 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
24		simprl 535	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → 𝑔 Fn 𝑧)
25		fneq2 5465	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑧 → (𝑔 Fn 𝑤 ↔ 𝑔 Fn 𝑧))
26		raleq 2749	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑧 → (∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)) ↔ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))
27	25, 26	anbi12d 477	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → ((𝑔 Fn 𝑤 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))) ↔ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))))
28	27	rspcev 2929	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝑋 ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → ∃𝑤 ∈ 𝑋 (𝑔 Fn 𝑤 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))
29	8, 28	sylan 283	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → ∃𝑤 ∈ 𝑋 (𝑔 Fn 𝑤 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))
30		vex 2824	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
31	12	tfr1onlem3ag 6598	. . . . . . . . . . 11 ⊢ (𝑔 ∈ V → (𝑔 ∈ 𝐴 ↔ ∃𝑤 ∈ 𝑋 (𝑔 Fn 𝑤 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))))
32	30, 31	ax-mp 5	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝐴 ↔ ∃𝑤 ∈ 𝑋 (𝑔 Fn 𝑤 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))
33	29, 32	sylibr 134	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → 𝑔 ∈ 𝐴)
34	9, 16, 17, 19, 12, 20, 21, 23, 24, 33	tfr1onlemsucaccv 6602	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ 𝐴)
35		vex 2824	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
36		fneq2 5465	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → (𝑔 Fn 𝑥 ↔ 𝑔 Fn 𝑧))
37	36	imbi1d 231	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → ((𝑔 Fn 𝑥 → (𝐺‘𝑔) ∈ V) ↔ (𝑔 Fn 𝑧 → (𝐺‘𝑔) ∈ V)))
38	11	3expia 1236	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V))
39	38	alrimiv 1927	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑓(𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V))
40		fneq1 5464	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝑥 ↔ 𝑔 Fn 𝑥))
41		fveq2 5690	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = 𝑔 → (𝐺‘𝑓) = (𝐺‘𝑔))
42	41	eleq1d 2307	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = 𝑔 → ((𝐺‘𝑓) ∈ V ↔ (𝐺‘𝑔) ∈ V))
43	40, 42	imbi12d 234	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = 𝑔 → ((𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V) ↔ (𝑔 Fn 𝑥 → (𝐺‘𝑔) ∈ V)))
44	43	spv 1913	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑓(𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V) → (𝑔 Fn 𝑥 → (𝐺‘𝑔) ∈ V))
45	39, 44	syl 14	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑔 Fn 𝑥 → (𝐺‘𝑔) ∈ V))
46	45	ralrimiva 2623	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 (𝑔 Fn 𝑥 → (𝐺‘𝑔) ∈ V))
47	46	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑌) → ∀𝑥 ∈ 𝑋 (𝑔 Fn 𝑥 → (𝐺‘𝑔) ∈ V))
48	37, 47, 8	rspcdva 2934	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑌) → (𝑔 Fn 𝑧 → (𝐺‘𝑔) ∈ V))
49	48	imp 124	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ 𝑔 Fn 𝑧) → (𝐺‘𝑔) ∈ V)
50	24, 49	syldan 282	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → (𝐺‘𝑔) ∈ V)
51		opexg 4363	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ V ∧ (𝐺‘𝑔) ∈ V) → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ V)
52	35, 50, 51	sylancr 418	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ V)
53		snidg 3734	. . . . . . . . . 10 ⊢ (⟨𝑧, (𝐺‘𝑔)⟩ ∈ V → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ {⟨𝑧, (𝐺‘𝑔)⟩})
54		elun2 3397	. . . . . . . . . 10 ⊢ (⟨𝑧, (𝐺‘𝑔)⟩ ∈ {⟨𝑧, (𝐺‘𝑔)⟩} → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))
55	52, 53, 54	3syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))
56		opeldmg 4981	. . . . . . . . . 10 ⊢ ((𝑧 ∈ V ∧ (𝐺‘𝑔) ∈ V) → (⟨𝑧, (𝐺‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})))
57	35, 50, 56	sylancr 418	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → (⟨𝑧, (𝐺‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})))
58	55, 57	mpd 13	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))
59		dmeq 4976	. . . . . . . . . 10 ⊢ (ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) → dom ℎ = dom (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))
60	59	eleq2d 2308	. . . . . . . . 9 ⊢ (ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) → (𝑧 ∈ dom ℎ ↔ 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})))
61	60	rspcev 2929	. . . . . . . 8 ⊢ (((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ 𝐴 ∧ 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})) → ∃ℎ ∈ 𝐴 𝑧 ∈ dom ℎ)
62	34, 58, 61	syl2anc 415	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → ∃ℎ ∈ 𝐴 𝑧 ∈ dom ℎ)
63	15, 62	exlimddv 1954	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑌) → ∃ℎ ∈ 𝐴 𝑧 ∈ dom ℎ)
64		eliun 4011	. . . . . 6 ⊢ (𝑧 ∈ ∪ ℎ ∈ 𝐴 dom ℎ ↔ ∃ℎ ∈ 𝐴 𝑧 ∈ dom ℎ)
65	63, 64	sylibr 134	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑌) → 𝑧 ∈ ∪ ℎ ∈ 𝐴 dom ℎ)
66	65	ex 115	. . . 4 ⊢ (𝜑 → (𝑧 ∈ 𝑌 → 𝑧 ∈ ∪ ℎ ∈ 𝐴 dom ℎ))
67	66	ssrdv 3254	. . 3 ⊢ (𝜑 → 𝑌 ⊆ ∪ ℎ ∈ 𝐴 dom ℎ)
68		dmuni 4986	. . . 4 ⊢ dom ∪ 𝐴 = ∪ ℎ ∈ 𝐴 dom ℎ
69	12, 1	tfr1onlemssrecs 6600	. . . . 5 ⊢ (𝜑 → ∪ 𝐴 ⊆ recs(𝐺))
70		dmss 4975	. . . . 5 ⊢ (∪ 𝐴 ⊆ recs(𝐺) → dom ∪ 𝐴 ⊆ dom recs(𝐺))
71	69, 70	syl 14	. . . 4 ⊢ (𝜑 → dom ∪ 𝐴 ⊆ dom recs(𝐺))
72	68, 71	eqsstrrid 3295	. . 3 ⊢ (𝜑 → ∪ ℎ ∈ 𝐴 dom ℎ ⊆ dom recs(𝐺))
73	67, 72	sstrd 3258	. 2 ⊢ (𝜑 → 𝑌 ⊆ dom recs(𝐺))
74	9	dmeqi 4977	. 2 ⊢ dom 𝐹 = dom recs(𝐺)
75	73, 74	sseqtrrdi 3297	1 ⊢ (𝜑 → 𝑌 ⊆ dom 𝐹)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009  ∀wal 1400   = wceq 1402  ∃wex 1545   ∈ wcel 2209  {cab 2224  ∀wral 2528  ∃wrex 2529  Vcvv 2821   ∪ cun 3218   ⊆ wss 3220  {csn 3705  ⟨cop 3708  ∪ cuni 3930  ∪ ciun 4007  Ord word 4502  suc csuc 4505  dom cdm 4769   ↾ cres 4771  Fun wfun 5366   Fn wfn 5367  ‘cfv 5372  recscrecs 6565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-recs 6566

This theorem is referenced by:  tfr1on  6611