Theorem wfrlem15 7969

Description: Lemma for well-founded recursion. When 𝑧 is 𝑅 minimal, 𝐶 is an acceptable function. This step is where the Axiom of Replacement becomes required. (Contributed by Scott Fenton, 21-Apr-2011.)

Hypotheses

Ref	Expression
wfrlem13.1	⊢ 𝑅 We 𝐴
wfrlem13.2	⊢ 𝑅 Se 𝐴
wfrlem13.3	⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
wfrlem13.4	⊢ 𝐶 = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})

Assertion

Ref	Expression
wfrlem15	⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → 𝐶 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})

Distinct variable groups:   𝐴,𝑓,𝑥,𝑦,𝑧   𝑓,𝐹,𝑥,𝑦,𝑧   𝑓,𝐺,𝑥,𝑦   𝑅,𝑓,𝑥,𝑦,𝑧   𝐶,𝑓,𝑥,𝑦

Allowed substitution hints:   𝐶(𝑧)   𝐺(𝑧)

Proof of Theorem wfrlem15

Step	Hyp	Ref	Expression
1		wfrlem13.1	. . . . 5 ⊢ 𝑅 We 𝐴
2		wfrlem13.2	. . . . 5 ⊢ 𝑅 Se 𝐴
3		wfrlem13.3	. . . . 5 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
4		wfrlem13.4	. . . . 5 ⊢ 𝐶 = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
5	1, 2, 3, 4	wfrlem13 7967	. . . 4 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝐶 Fn (dom 𝐹 ∪ {𝑧}))
6	5	adantr 483	. . 3 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → 𝐶 Fn (dom 𝐹 ∪ {𝑧}))
7	1, 3	wfrlem10 7964	. . . 4 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → Pred(𝑅, 𝐴, 𝑧) = dom 𝐹)
8		eldifi 4103	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝑧 ∈ 𝐴)
9		setlikespec 6169	. . . . . 6 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑧) ∈ V)
10	8, 2, 9	sylancl 588	. . . . 5 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → Pred(𝑅, 𝐴, 𝑧) ∈ V)
11	10	adantr 483	. . . 4 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → Pred(𝑅, 𝐴, 𝑧) ∈ V)
12	7, 11	eqeltrrd 2914	. . 3 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → dom 𝐹 ∈ V)
13		snex 5332	. . . . 5 ⊢ {𝑧} ∈ V
14		unexg 7472	. . . . 5 ⊢ ((dom 𝐹 ∈ V ∧ {𝑧} ∈ V) → (dom 𝐹 ∪ {𝑧}) ∈ V)
15	13, 14	mpan2 689	. . . 4 ⊢ (dom 𝐹 ∈ V → (dom 𝐹 ∪ {𝑧}) ∈ V)
16		fnex 6980	. . . 4 ⊢ ((𝐶 Fn (dom 𝐹 ∪ {𝑧}) ∧ (dom 𝐹 ∪ {𝑧}) ∈ V) → 𝐶 ∈ V)
17	15, 16	sylan2 594	. . 3 ⊢ ((𝐶 Fn (dom 𝐹 ∪ {𝑧}) ∧ dom 𝐹 ∈ V) → 𝐶 ∈ V)
18	6, 12, 17	syl2anc 586	. 2 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → 𝐶 ∈ V)
19	12, 13, 14	sylancl 588	. . 3 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → (dom 𝐹 ∪ {𝑧}) ∈ V)
20	3	wfrdmss 7961	. . . . . . 7 ⊢ dom 𝐹 ⊆ 𝐴
21	8	snssd 4742	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → {𝑧} ⊆ 𝐴)
22		unss 4160	. . . . . . . 8 ⊢ ((dom 𝐹 ⊆ 𝐴 ∧ {𝑧} ⊆ 𝐴) ↔ (dom 𝐹 ∪ {𝑧}) ⊆ 𝐴)
23	22	biimpi 218	. . . . . . 7 ⊢ ((dom 𝐹 ⊆ 𝐴 ∧ {𝑧} ⊆ 𝐴) → (dom 𝐹 ∪ {𝑧}) ⊆ 𝐴)
24	20, 21, 23	sylancr 589	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (dom 𝐹 ∪ {𝑧}) ⊆ 𝐴)
25	24	adantr 483	. . . . 5 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → (dom 𝐹 ∪ {𝑧}) ⊆ 𝐴)
26		elun 4125	. . . . . . . 8 ⊢ (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) ↔ (𝑦 ∈ dom 𝐹 ∨ 𝑦 ∈ {𝑧}))
27		velsn 4583	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑧} ↔ 𝑦 = 𝑧)
28	27	orbi2i 909	. . . . . . . 8 ⊢ ((𝑦 ∈ dom 𝐹 ∨ 𝑦 ∈ {𝑧}) ↔ (𝑦 ∈ dom 𝐹 ∨ 𝑦 = 𝑧))
29	26, 28	bitri 277	. . . . . . 7 ⊢ (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) ↔ (𝑦 ∈ dom 𝐹 ∨ 𝑦 = 𝑧))
30	3	wfrdmcl 7963	. . . . . . . . . 10 ⊢ (𝑦 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝐹)
31		ssun3 4150	. . . . . . . . . 10 ⊢ (Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝐹 → Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧}))
32	30, 31	syl 17	. . . . . . . . 9 ⊢ (𝑦 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧}))
33	32	a1i 11	. . . . . . . 8 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → (𝑦 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧})))
34		ssun1 4148	. . . . . . . . . 10 ⊢ dom 𝐹 ⊆ (dom 𝐹 ∪ {𝑧})
35	7, 34	eqsstrdi 4021	. . . . . . . . 9 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → Pred(𝑅, 𝐴, 𝑧) ⊆ (dom 𝐹 ∪ {𝑧}))
36		predeq3 6152	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑧))
37	36	sseq1d 3998	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧}) ↔ Pred(𝑅, 𝐴, 𝑧) ⊆ (dom 𝐹 ∪ {𝑧})))
38	35, 37	syl5ibrcom 249	. . . . . . . 8 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → (𝑦 = 𝑧 → Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧})))
39	33, 38	jaod 855	. . . . . . 7 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → ((𝑦 ∈ dom 𝐹 ∨ 𝑦 = 𝑧) → Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧})))
40	29, 39	syl5bi 244	. . . . . 6 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) → Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧})))
41	40	ralrimiv 3181	. . . . 5 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧}))
42	25, 41	jca 514	. . . 4 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → ((dom 𝐹 ∪ {𝑧}) ⊆ 𝐴 ∧ ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧})))
43	1, 2, 3, 4	wfrlem14 7968	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) → (𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
44	43	ralrimiv 3181	. . . . 5 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})(𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))))
45	44	adantr 483	. . . 4 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})(𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))))
46	6, 42, 45	3jca 1124	. . 3 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → (𝐶 Fn (dom 𝐹 ∪ {𝑧}) ∧ ((dom 𝐹 ∪ {𝑧}) ⊆ 𝐴 ∧ ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧})) ∧ ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})(𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
47		fneq2 6445	. . . 4 ⊢ (𝑥 = (dom 𝐹 ∪ {𝑧}) → (𝐶 Fn 𝑥 ↔ 𝐶 Fn (dom 𝐹 ∪ {𝑧})))
48		sseq1 3992	. . . . 5 ⊢ (𝑥 = (dom 𝐹 ∪ {𝑧}) → (𝑥 ⊆ 𝐴 ↔ (dom 𝐹 ∪ {𝑧}) ⊆ 𝐴))
49		sseq2 3993	. . . . . 6 ⊢ (𝑥 = (dom 𝐹 ∪ {𝑧}) → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧})))
50	49	raleqbi1dv 3403	. . . . 5 ⊢ (𝑥 = (dom 𝐹 ∪ {𝑧}) → (∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧})))
51	48, 50	anbi12d 632	. . . 4 ⊢ (𝑥 = (dom 𝐹 ∪ {𝑧}) → ((𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ↔ ((dom 𝐹 ∪ {𝑧}) ⊆ 𝐴 ∧ ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧}))))
52		raleq 3405	. . . 4 ⊢ (𝑥 = (dom 𝐹 ∪ {𝑧}) → (∀𝑦 ∈ 𝑥 (𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})(𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
53	47, 51, 52	3anbi123d 1432	. . 3 ⊢ (𝑥 = (dom 𝐹 ∪ {𝑧}) → ((𝐶 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ (𝐶 Fn (dom 𝐹 ∪ {𝑧}) ∧ ((dom 𝐹 ∪ {𝑧}) ⊆ 𝐴 ∧ ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧})) ∧ ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})(𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))))))
54	19, 46, 53	spcedv 3599	. 2 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → ∃𝑥(𝐶 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
55		fneq1 6444	. . . 4 ⊢ (𝑓 = 𝐶 → (𝑓 Fn 𝑥 ↔ 𝐶 Fn 𝑥))
56		fveq1 6669	. . . . . 6 ⊢ (𝑓 = 𝐶 → (𝑓‘𝑦) = (𝐶‘𝑦))
57		reseq1 5847	. . . . . . 7 ⊢ (𝑓 = 𝐶 → (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))
58	57	fveq2d 6674	. . . . . 6 ⊢ (𝑓 = 𝐶 → (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))))
59	56, 58	eqeq12d 2837	. . . . 5 ⊢ (𝑓 = 𝐶 → ((𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
60	59	ralbidv 3197	. . . 4 ⊢ (𝑓 = 𝐶 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑦 ∈ 𝑥 (𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
61	55, 60	3anbi13d 1434	. . 3 ⊢ (𝑓 = 𝐶 → ((𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ (𝐶 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))))))
62	61	exbidv 1922	. 2 ⊢ (𝑓 = 𝐶 → (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ ∃𝑥(𝐶 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))))))
63	18, 54, 62	elabd 3669	1 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → 𝐶 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})

Syntax hints:   → wi 4   ∧ wa 398   ∨ wo 843   ∧ w3a 1083   = wceq 1537  ∃wex 1780   ∈ wcel 2114  {cab 2799  ∀wral 3138  Vcvv 3494   ∖ cdif 3933   ∪ cun 3934   ⊆ wss 3936  ∅c0 4291  {csn 4567  ⟨cop 4573   Se wse 5512   We wwe 5513  dom cdm 5555   ↾ cres 5557  Predcpred 6147   Fn wfn 6350  ‘cfv 6355  wrecscwrecs 7946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-wrecs 7947

This theorem is referenced by:  wfrlem16  7970