Theorem tfrlem9 8307

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 5842	. . 3 ⊢ (𝐵 ∈ dom recs(𝐹) → (𝐵 ∈ dom recs(𝐹) ↔ ∃𝑧⟨𝐵, 𝑧⟩ ∈ recs(𝐹)))
2	1	ibi 267	. 2 ⊢ (𝐵 ∈ dom recs(𝐹) → ∃𝑧⟨𝐵, 𝑧⟩ ∈ recs(𝐹))
3		dfrecs3 8295	. . . . . 6 ⊢ recs(𝐹) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
4	3	eleq2i 2820	. . . . 5 ⊢ (⟨𝐵, 𝑧⟩ ∈ recs(𝐹) ↔ ⟨𝐵, 𝑧⟩ ∈ ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))})
5		eluniab 4872	. . . . 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 275	. . . 4 ⊢ (⟨𝐵, 𝑧⟩ ∈ recs(𝐹) ↔ ∃𝑓(⟨𝐵, 𝑧⟩ ∈ 𝑓 ∧ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))))
7		fnop 6591	. . . . . . . . . . . . . 14 ⊢ ((𝑓 Fn 𝑥 ∧ ⟨𝐵, 𝑧⟩ ∈ 𝑓) → 𝐵 ∈ 𝑥)
8		rspe 3219	. . . . . . . . . . . . . . . 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 2871	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ 𝐴 ↔ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
11		elssuni 4888	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ 𝐴 → 𝑓 ⊆ ∪ 𝐴)
12	9	recsfval 8303	. . . . . . . . . . . . . . . . . 18 ⊢ recs(𝐹) = ∪ 𝐴
13	11, 12	sseqtrrdi 3977	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ 𝐴 → 𝑓 ⊆ recs(𝐹))
14	10, 13	sylbir 235	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))) → 𝑓 ⊆ recs(𝐹))
15	8, 14	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))) → 𝑓 ⊆ recs(𝐹))
16		fveq2 6822	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝐵 → (𝑓‘𝑦) = (𝑓‘𝐵))
17		reseq2 5925	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝐵 → (𝑓 ↾ 𝑦) = (𝑓 ↾ 𝐵))
18	17	fveq2d 6826	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝐵 → (𝐹‘(𝑓 ↾ 𝑦)) = (𝐹‘(𝑓 ↾ 𝐵)))
19	16, 18	eqeq12d 2745	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝐵 → ((𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ (𝑓‘𝐵) = (𝐹‘(𝑓 ↾ 𝐵))))
20	19	rspcv 3573	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) → (𝑓‘𝐵) = (𝐹‘(𝑓 ↾ 𝐵))))
21		fndm 6585	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 Fn 𝑥 → dom 𝑓 = 𝑥)
22	21	eleq2d 2814	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 Fn 𝑥 → (𝐵 ∈ dom 𝑓 ↔ 𝐵 ∈ 𝑥))
23	9	tfrlem7 8305	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ Fun recs(𝐹)
24		funssfv 6843	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((Fun recs(𝐹) ∧ 𝑓 ⊆ recs(𝐹) ∧ 𝐵 ∈ dom 𝑓) → (recs(𝐹)‘𝐵) = (𝑓‘𝐵))
25	23, 24	mp3an1 1450	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓 ⊆ recs(𝐹) ∧ 𝐵 ∈ dom 𝑓) → (recs(𝐹)‘𝐵) = (𝑓‘𝐵))
26	25	adantrl 716	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓 ⊆ recs(𝐹) ∧ ((𝑓 Fn 𝑥 ∧ 𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓)) → (recs(𝐹)‘𝐵) = (𝑓‘𝐵))
27	21	eleq1d 2813	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑓 Fn 𝑥 → (dom 𝑓 ∈ On ↔ 𝑥 ∈ On))
28		onelss 6349	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (dom 𝑓 ∈ On → (𝐵 ∈ dom 𝑓 → 𝐵 ⊆ dom 𝑓))
29	27, 28	biimtrrdi 254	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑓 Fn 𝑥 → (𝑥 ∈ On → (𝐵 ∈ dom 𝑓 → 𝐵 ⊆ dom 𝑓)))
30	29	imp31 417	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 Fn 𝑥 ∧ 𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓) → 𝐵 ⊆ dom 𝑓)
31		fun2ssres 6527	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((Fun recs(𝐹) ∧ 𝑓 ⊆ recs(𝐹) ∧ 𝐵 ⊆ dom 𝑓) → (recs(𝐹) ↾ 𝐵) = (𝑓 ↾ 𝐵))
32	31	fveq2d 6826	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((Fun recs(𝐹) ∧ 𝑓 ⊆ recs(𝐹) ∧ 𝐵 ⊆ dom 𝑓) → (𝐹‘(recs(𝐹) ↾ 𝐵)) = (𝐹‘(𝑓 ↾ 𝐵)))
33	23, 32	mp3an1 1450	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓 ⊆ recs(𝐹) ∧ 𝐵 ⊆ dom 𝑓) → (𝐹‘(recs(𝐹) ↾ 𝐵)) = (𝐹‘(𝑓 ↾ 𝐵)))
34	30, 33	sylan2 593	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓 ⊆ recs(𝐹) ∧ ((𝑓 Fn 𝑥 ∧ 𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓)) → (𝐹‘(recs(𝐹) ↾ 𝐵)) = (𝐹‘(𝑓 ↾ 𝐵)))
35	26, 34	eqeq12d 2745	. . . . . . . . . . . . . . . . . . . . . . . . 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 810	. . . . . . . . . . . . . . . . . . . . . . . 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 89	. . . . . . . . . . . . . . . . . . . . . . 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 419	. . . . . . . . . . . . . . . . . . . . . 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 91	. . . . . . . . . . . . . . . . . . . . 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 95	. . . . . . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . . . . . 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 89	. . . . . . . . . . . . . . . . . 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 47	. . . . . . . . . . . . . . . . 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 95	. . . . . . . . . . . . . . . 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 424	. . . . . . . . . . . . . . 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 45	. . . . . . . . . . . . . 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 17	. . . . . . . . . . . . 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 433	. . . . . . . . . . . 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 412	. . . . . . . . . . 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 94	. . . . . . . . . 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 52	. . . . . . . . 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 89	. . . . . . . 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 422	. . . . . . 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 3128	. . . . . 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 406	. . . . 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 1930	. . . 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 217	. . 3 ⊢ (⟨𝐵, 𝑧⟩ ∈ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
58	57	exlimiv 1930	. 2 ⊢ (∃𝑧⟨𝐵, 𝑧⟩ ∈ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
59	2, 58	syl 17	1 ⊢ (𝐵 ∈ dom recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707  ∀wral 3044  ∃wrex 3053   ⊆ wss 3903  ⟨cop 4583  ∪ cuni 4858  dom cdm 5619   ↾ cres 5621  Oncon0 6307  Fun wfun 6476   Fn wfn 6477  ‘cfv 6482  recscrecs 8293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-fo 6488  df-fv 6490  df-ov 7352  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294

This theorem is referenced by:  tfrlem11  8310  tfr2a  8317