Theorem tfrlem9 8385

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 5900	. . 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 8372	. . . . . 6 ⊢ recs(𝐹) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
4	3	eleq2i 2826	. . . . 5 ⊢ (⟨𝐵, 𝑧⟩ ∈ recs(𝐹) ↔ ⟨𝐵, 𝑧⟩ ∈ ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))})
5		eluniab 4924	. . . . 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 6659	. . . . . . . . . . . . . 14 ⊢ ((𝑓 Fn 𝑥 ∧ ⟨𝐵, 𝑧⟩ ∈ 𝑓) → 𝐵 ∈ 𝑥)
8		rspe 3247	. . . . . . . . . . . . . . . 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 2878	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ 𝐴 ↔ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))))
11		elssuni 4942	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 ∈ 𝐴 → 𝑓 ⊆ ∪ 𝐴)
12	9	recsfval 8381	. . . . . . . . . . . . . . . . . 18 ⊢ recs(𝐹) = ∪ 𝐴
13	11, 12	sseqtrrdi 4034	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 ∈ 𝐴 → 𝑓 ⊆ recs(𝐹))
14	10, 13	sylbir 234	. . . . . . . . . . . . . . . 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 6892	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝐵 → (𝑓‘𝑦) = (𝑓‘𝐵))
17		reseq2 5977	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝐵 → (𝑓 ↾ 𝑦) = (𝑓 ↾ 𝐵))
18	17	fveq2d 6896	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝐵 → (𝐹‘(𝑓 ↾ 𝑦)) = (𝐹‘(𝑓 ↾ 𝐵)))
19	16, 18	eqeq12d 2749	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝐵 → ((𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ (𝑓‘𝐵) = (𝐹‘(𝑓 ↾ 𝐵))))
20	19	rspcv 3609	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∈ 𝑥 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) → (𝑓‘𝐵) = (𝐹‘(𝑓 ↾ 𝐵))))
21		fndm 6653	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓 Fn 𝑥 → dom 𝑓 = 𝑥)
22	21	eleq2d 2820	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 Fn 𝑥 → (𝐵 ∈ dom 𝑓 ↔ 𝐵 ∈ 𝑥))
23	9	tfrlem7 8383	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ Fun recs(𝐹)
24		funssfv 6913	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((Fun recs(𝐹) ∧ 𝑓 ⊆ recs(𝐹) ∧ 𝐵 ∈ dom 𝑓) → (recs(𝐹)‘𝐵) = (𝑓‘𝐵))
25	23, 24	mp3an1 1449	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓 ⊆ recs(𝐹) ∧ 𝐵 ∈ dom 𝑓) → (recs(𝐹)‘𝐵) = (𝑓‘𝐵))
26	25	adantrl 715	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓 ⊆ recs(𝐹) ∧ ((𝑓 Fn 𝑥 ∧ 𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓)) → (recs(𝐹)‘𝐵) = (𝑓‘𝐵))
27	21	eleq1d 2819	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑓 Fn 𝑥 → (dom 𝑓 ∈ On ↔ 𝑥 ∈ On))
28		onelss 6407	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (dom 𝑓 ∈ On → (𝐵 ∈ dom 𝑓 → 𝐵 ⊆ dom 𝑓))
29	27, 28	syl6bir 254	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑓 Fn 𝑥 → (𝑥 ∈ On → (𝐵 ∈ dom 𝑓 → 𝐵 ⊆ dom 𝑓)))
30	29	imp31 419	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑓 Fn 𝑥 ∧ 𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓) → 𝐵 ⊆ dom 𝑓)
31		fun2ssres 6594	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((Fun recs(𝐹) ∧ 𝑓 ⊆ recs(𝐹) ∧ 𝐵 ⊆ dom 𝑓) → (recs(𝐹) ↾ 𝐵) = (𝑓 ↾ 𝐵))
32	31	fveq2d 6896	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((Fun recs(𝐹) ∧ 𝑓 ⊆ recs(𝐹) ∧ 𝐵 ⊆ dom 𝑓) → (𝐹‘(recs(𝐹) ↾ 𝐵)) = (𝐹‘(𝑓 ↾ 𝐵)))
33	23, 32	mp3an1 1449	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑓 ⊆ recs(𝐹) ∧ 𝐵 ⊆ dom 𝑓) → (𝐹‘(recs(𝐹) ↾ 𝐵)) = (𝐹‘(𝑓 ↾ 𝐵)))
34	30, 33	sylan2 594	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑓 ⊆ recs(𝐹) ∧ ((𝑓 Fn 𝑥 ∧ 𝑥 ∈ On) ∧ 𝐵 ∈ dom 𝑓)) → (𝐹‘(recs(𝐹) ↾ 𝐵)) = (𝐹‘(𝑓 ↾ 𝐵)))
35	26, 34	eqeq12d 2749	. . . . . . . . . . . . . . . . . . . . . . . . 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 421	. . . . . . . . . . . . . . . . . . . . . 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 426	. . . . . . . . . . . . . . 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 435	. . . . . . . . . . . 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 414	. . . . . . . . . . 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 424	. . . . . . 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 3154	. . . . . 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 408	. . . . 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 1934	. . . 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 216	. . 3 ⊢ (⟨𝐵, 𝑧⟩ ∈ recs(𝐹) → (recs(𝐹)‘𝐵) = (𝐹‘(recs(𝐹) ↾ 𝐵)))
58	57	exlimiv 1934	. 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 397   ∧ w3a 1088   = wceq 1542  ∃wex 1782   ∈ wcel 2107  {cab 2710  ∀wral 3062  ∃wrex 3071   ⊆ wss 3949  ⟨cop 4635  ∪ cuni 4909  dom cdm 5677   ↾ cres 5679  Oncon0 6365  Fun wfun 6538   Fn wfn 6539  ‘cfv 6544  recscrecs 8370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-ov 7412  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371

This theorem is referenced by:  tfrlem11  8388  tfr2a  8395