Theorem tfrlem9 8379

Description: Lemma for transfinite recursion. Here we compute the value of recs (the union of all acceptable functions). (Contributed by NM, 17-Aug-1994.)

Hypothesis

Ref	Expression
tfrlem.1	⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}

Assertion

Ref	Expression
tfrlem9	⊢ (𝐵 ∈ dom recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))

Distinct variable groups:   𝑥,𝑓,𝑦,𝐵   𝑓,𝐹,𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥, 𝑦, 𝑓)

Proof of Theorem tfrlem9

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldm2g 5885	. . 3 ⊢ (𝐵 ∈ dom recs(𝐹) → (𝐵 ∈ dom recs(𝐹) ↔ ∃𝑧⟨𝐵, 𝑧⟩ ∈ recs(𝐹)))
2	1	ibi 270	. 2 ⊢ (𝐵 ∈ dom recs(𝐹) → ∃𝑧⟨𝐵, 𝑧⟩ ∈ recs(𝐹))
3		dfrecs3 8366	. . . . . 6 ⊢ recs(𝐹) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
4	3	eleq2i 2852	. . . . 5 ⊢ (⟨𝐵, 𝑧⟩ ∈ recs(𝐹) ↔ ⟨𝐵, 𝑧⟩ ∈ ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))})
5		eluniab 4881	. . . . 5 ⊢ (⟨𝐵, 𝑧⟩ ∈ ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} ↔ ∃𝑓(⟨𝐵, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))))
6	4, 5	bitri 278	. . . 4 ⊢ (⟨𝐵, 𝑧⟩ ∈ recs(𝐹) ↔ ∃𝑓(⟨𝐵, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))))
7		fnop 6644	. . . . . . . . . . . . . 14 ⊢ ((𝑓 Fn 𝑥 ∧ ⟨𝐵, 𝑧⟩ ∈ 𝑓) → 𝐵 ∈ 𝑥)
8		rspe 3252	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))) → ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
9		tfrlem.1	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
10	9	eqabri 2902	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ 𝐴 ↔ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
11		elssuni 4899	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ 𝐴 → 𝑓 ⊆ ∪ 𝐴)
12	9	recsfval 8374	. . . . . . . . . . . . . . . . . 18 ⊢ recs(𝐹) = ∪ 𝐴
13	11, 12	sseqtrrdi 3972	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ 𝐴 → 𝑓 ⊆ recs(𝐹))
14	10, 13	sylbir 238	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))) → 𝑓 ⊆ recs(𝐹))
15	8, 14	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))) → 𝑓 ⊆ recs(𝐹))
16		fveq2 6881	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝐵 → (𝑓‘𝑦) = (𝑓‘𝐵))
17		reseq2 5969	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝐵 → (𝑓 ↾ 𝑦) = (𝑓 ↾ 𝐵))
18	17	fveq2d 6885	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝐵 → (𝐹‘(𝑓 ↾ 𝑦)) = (𝐹‘(𝑓 ↾ 𝐵)))
19	16, 18	eqeq12d 2776	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝐵 → ((𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ (𝑓‘𝐵) = (𝐹‘(𝑓 ↾ 𝐵))))
20	19	rspcv 3572	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) → (𝑓‘𝐵) = (𝐹‘(𝑓 ↾ 𝐵))))
21		fndm 6638	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 Fn 𝑥 → dom 𝑓 = 𝑥)
22	21	eleq2d 2846	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 Fn 𝑥 → (𝐵 ∈ dom 𝑓 ↔ 𝐵 ∈ 𝑥))
23	9	tfrlem7 8377	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ Fun recs(𝐹)
24		funssfv 6902	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((Fun recs(𝐹) ∧ 𝑓 ⊆ recs(𝐹) ∧ 𝐵 ∈ dom 𝑓) → (recs(𝐹)‘𝐵) = (𝑓‘𝐵))
25	23, 24	mp3an1 1477	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓 ⊆ recs(𝐹) ∧ 𝐵 ∈ dom 𝑓) → (recs(𝐹)‘𝐵) = (𝑓‘𝐵))
26	25	adantrl 729	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓 ⊆ recs(𝐹) ∧ ((𝑓 Fn 𝑥 ∧ 𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓)) → (recs(𝐹)‘𝐵) = (𝑓‘𝐵))
27	21	eleq1d 2845	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑓 Fn 𝑥 → (dom 𝑓 ∈ On ↔ 𝑥 ∈ On))
28		onelss 6402	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (dom 𝑓 ∈ On → (𝐵 ∈ dom 𝑓 → 𝐵 ⊆ dom 𝑓))
29	27, 28	biimtrrdi 257	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑓 Fn 𝑥 → (𝑥 ∈ On → (𝐵 ∈ dom 𝑓 → 𝐵 ⊆ dom 𝑓)))
30	29	imp31 423	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 Fn 𝑥 ∧ 𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓) → 𝐵 ⊆ dom 𝑓)
31		fun2ssres 6581	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((Fun recs(𝐹) ∧ 𝑓 ⊆ recs(𝐹) ∧ 𝐵 ⊆ dom 𝑓) → (recs(𝐹) ↾ 𝐵) = (𝑓 ↾ 𝐵))
32	31	fveq2d 6885	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((Fun recs(𝐹) ∧ 𝑓 ⊆ recs(𝐹) ∧ 𝐵 ⊆ dom 𝑓) → (𝐹‘(recs(𝐹) ↾ 𝐵)) = (𝐹‘(𝑓 ↾ 𝐵)))
33	23, 32	mp3an1 1477	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓 ⊆ recs(𝐹) ∧ 𝐵 ⊆ dom 𝑓) → (𝐹‘(recs(𝐹) ↾ 𝐵)) = (𝐹‘(𝑓 ↾ 𝐵)))
34	30, 33	sylan2 605	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓 ⊆ recs(𝐹) ∧ ((𝑓 Fn 𝑥 ∧ 𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓)) → (𝐹‘(recs(𝐹) ↾ 𝐵)) = (𝐹‘(𝑓 ↾ 𝐵)))
35	26, 34	eqeq12d 2776	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 ⊆ recs(𝐹) ∧ ((𝑓 Fn 𝑥 ∧ 𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓)) → ((recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)) ↔ (𝑓‘𝐵) = (𝐹‘(𝑓 ↾ 𝐵))))
36	35	exbiri 823	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 ⊆ recs(𝐹) → (((𝑓 Fn 𝑥 ∧ 𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓) → ((𝑓‘𝐵) = (𝐹‘(𝑓 ↾ 𝐵)) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))
37	36	com3l 90	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑓 Fn 𝑥 ∧ 𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓) → ((𝑓‘𝐵) = (𝐹‘(𝑓 ↾ 𝐵)) → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))
38	37	exp31 425	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓 Fn 𝑥 → (𝑥 ∈ On → (𝐵 ∈ dom 𝑓 → ((𝑓‘𝐵) = (𝐹‘(𝑓 ↾ 𝐵)) → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
39	38	com34 92	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 Fn 𝑥 → (𝑥 ∈ On → ((𝑓‘𝐵) = (𝐹‘(𝑓 ↾ 𝐵)) → (𝐵 ∈ dom 𝑓 → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
40	39	com24 96	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 Fn 𝑥 → (𝐵 ∈ dom 𝑓 → ((𝑓‘𝐵) = (𝐹‘(𝑓 ↾ 𝐵)) → (𝑥 ∈ On → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
41	22, 40	sylbird 263	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 Fn 𝑥 → (𝐵 ∈ 𝑥 → ((𝑓‘𝐵) = (𝐹‘(𝑓 ↾ 𝐵)) → (𝑥 ∈ On → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
42	41	com3l 90	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ 𝑥 → ((𝑓‘𝐵) = (𝐹‘(𝑓 ↾ 𝐵)) → (𝑓 Fn 𝑥 → (𝑥 ∈ On → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
43	20, 42	syld 48	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) → (𝑓 Fn 𝑥 → (𝑥 ∈ On → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
44	43	com24 96	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ 𝑥 → (𝑥 ∈ On → (𝑓 Fn 𝑥 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
45	44	imp4d 430	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ 𝑥 → ((𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))) → (𝑓 ⊆ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))
46	15, 45	mpdi 46	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ 𝑥 → ((𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵))))
47	7, 46	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝑓 Fn 𝑥 ∧ ⟨𝐵, 𝑧⟩ ∈ 𝑓) → ((𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵))))
48	47	exp4d 439	. . . . . . . . . . . 12 ⊢ ((𝑓 Fn 𝑥 ∧ ⟨𝐵, 𝑧⟩ ∈ 𝑓) → (𝑥 ∈ On → (𝑓 Fn 𝑥 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵))))))
49	48	ex 418	. . . . . . . . . . 11 ⊢ (𝑓 Fn 𝑥 → (⟨𝐵, 𝑧⟩ ∈ 𝑓 → (𝑥 ∈ On → (𝑓 Fn 𝑥 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
50	49	com4r 95	. . . . . . . . . 10 ⊢ (𝑓 Fn 𝑥 → (𝑓 Fn 𝑥 → (⟨𝐵, 𝑧⟩ ∈ 𝑓 → (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))))
51	50	pm2.43i 53	. . . . . . . . 9 ⊢ (𝑓 Fn 𝑥 → (⟨𝐵, 𝑧⟩ ∈ 𝑓 → (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵))))))
52	51	com3l 90	. . . . . . . 8 ⊢ (⟨𝐵, 𝑧⟩ ∈ 𝑓 → (𝑥 ∈ On → (𝑓 Fn 𝑥 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵))))))
53	52	imp4a 428	. . . . . . 7 ⊢ (⟨𝐵, 𝑧⟩ ∈ 𝑓 → (𝑥 ∈ On → ((𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))))
54	53	rexlimdv 3161	. . . . . 6 ⊢ (⟨𝐵, 𝑧⟩ ∈ 𝑓 → (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵))))
55	54	imp 412	. . . . 5 ⊢ ((⟨𝐵, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
56	55	exlimiv 1963	. . . 4 ⊢ (∃𝑓(⟨𝐵, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
57	6, 56	sylbi 220	. . 3 ⊢ (⟨𝐵, 𝑧⟩ ∈ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
58	57	exlimiv 1963	. 2 ⊢ (∃𝑧⟨𝐵, 𝑧⟩ ∈ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
59	2, 58	syl 18	1 ⊢ (𝐵 ∈ dom recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2738  ∀wral 3076  ∃wrex 3086   ⊆ wss 3899  ⟨cop 4590  ∪ cuni 4867  dom cdm 5655   ↾ cres 5657  Oncon0 6359  Fun wfun 6529   Fn wfn 6530  ‘cfv 6535  recscrecs 8364

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7742

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6301  df-ord 6362  df-on 6363  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-fo 6541  df-fv 6543  df-ov 7419  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8365

This theorem is used by:  tfrlem11  8382  tfr2a  8389