Theorem tfr1onlemres 6593

Description: Lemma for tfr1on 6594. 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 4514	. . . . . . . . 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 6592	. . . . . . . 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 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → Fun 𝐺)
17	1	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → Ord 𝑋)
18	11	3adant1r 1258	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋 ∧ 𝑓 Fn 𝑥) → (𝐺‘𝑓) ∈ V)
19	18	3adant1r 1258	. . . . . . . . 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 488	. . . . . . . . 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 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
23	22	adantlr 477	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
24		simprl 531	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → 𝑔 Fn 𝑧)
25		fneq2 5450	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑧 → (𝑔 Fn 𝑤 ↔ 𝑔 Fn 𝑧))
26		raleq 2743	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑧 → (∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)) ↔ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))
27	25, 26	anbi12d 473	. . . . . . . . . . . 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 2923	. . . . . . . . . . 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 2818	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
31	12	tfr1onlem3ag 6581	. . . . . . . . . . 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 6585	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ 𝐴)
35		vex 2818	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
36		fneq2 5450	. . . . . . . . . . . . . . 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 1232	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V))
39	38	alrimiv 1923	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑓(𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V))
40		fneq1 5449	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝑥 ↔ 𝑔 Fn 𝑥))
41		fveq2 5675	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = 𝑔 → (𝐺‘𝑓) = (𝐺‘𝑔))
42	41	eleq1d 2303	. . . . . . . . . . . . . . . . . . 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 1909	. . . . . . . . . . . . . . . . 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 2617	. . . . . . . . . . . . . . 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 2928	. . . . . . . . . . . . 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 4349	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ V ∧ (𝐺‘𝑔) ∈ V) → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ V)
52	35, 50, 51	sylancr 414	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ V)
53		snidg 3723	. . . . . . . . . 10 ⊢ (⟨𝑧, (𝐺‘𝑔)⟩ ∈ V → ⟨𝑧, (𝐺‘𝑔)⟩ ∈ {⟨𝑧, (𝐺‘𝑔)⟩})
54		elun2 3391	. . . . . . . . . 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 4966	. . . . . . . . . 10 ⊢ ((𝑧 ∈ V ∧ (𝐺‘𝑔) ∈ V) → (⟨𝑧, (𝐺‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})))
57	35, 50, 56	sylancr 414	. . . . . . . . 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 4961	. . . . . . . . . 10 ⊢ (ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) → dom ℎ = dom (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))
60	59	eleq2d 2304	. . . . . . . . 9 ⊢ (ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) → (𝑧 ∈ dom ℎ ↔ 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})))
61	60	rspcev 2923	. . . . . . . 8 ⊢ (((𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ 𝐴 ∧ 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})) → ∃ℎ ∈ 𝐴 𝑧 ∈ dom ℎ)
62	34, 58, 61	syl2anc 411	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑌) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → ∃ℎ ∈ 𝐴 𝑧 ∈ dom ℎ)
63	15, 62	exlimddv 1950	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑌) → ∃ℎ ∈ 𝐴 𝑧 ∈ dom ℎ)
64		eliun 4000	. . . . . 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 3248	. . 3 ⊢ (𝜑 → 𝑌 ⊆ ∪ ℎ ∈ 𝐴 dom ℎ)
68		dmuni 4971	. . . 4 ⊢ dom ∪ 𝐴 = ∪ ℎ ∈ 𝐴 dom ℎ
69	12, 1	tfr1onlemssrecs 6583	. . . . 5 ⊢ (𝜑 → ∪ 𝐴 ⊆ recs(𝐺))
70		dmss 4960	. . . . 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 3289	. . 3 ⊢ (𝜑 → ∪ ℎ ∈ 𝐴 dom ℎ ⊆ dom recs(𝐺))
73	67, 72	sstrd 3252	. 2 ⊢ (𝜑 → 𝑌 ⊆ dom recs(𝐺))
74	9	dmeqi 4962	. 2 ⊢ dom 𝐹 = dom recs(𝐺)
75	73, 74	sseqtrrdi 3291	1 ⊢ (𝜑 → 𝑌 ⊆ dom 𝐹)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005  ∀wal 1396   = wceq 1398  ∃wex 1541   ∈ wcel 2205  {cab 2220  ∀wral 2522  ∃wrex 2523  Vcvv 2815   ∪ cun 3212   ⊆ wss 3214  {csn 3694  ⟨cop 3697  ∪ cuni 3919  ∪ ciun 3996  Ord word 4488  suc csuc 4491  dom cdm 4754   ↾ cres 4756  Fun wfun 5351   Fn wfn 5352  ‘cfv 5357  recscrecs 6548

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-recs 6549

This theorem is referenced by:  tfr1on  6594