Theorem wfrlem15OLD 8125

Description: Lemma for well-ordered recursion. When 𝑧 is 𝑅 minimal, 𝐶 is an acceptable function. This step is where the Axiom of Replacement becomes required. Obsolete as of 18-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 21-Apr-2011.)

Hypotheses

Ref	Expression
wfrlem13OLD.1	⊢ 𝑅 We 𝐴
wfrlem13OLD.2	⊢ 𝑅 Se 𝐴
wfrlem13OLD.3	⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
wfrlem13OLD.4	⊢ 𝐶 = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})

Assertion

Ref	Expression
wfrlem15OLD	⊢ ((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 wfrlem15OLD

Step	Hyp	Ref	Expression
1		wfrlem13OLD.1	. . . . 5 ⊢ 𝑅 We 𝐴
2		wfrlem13OLD.2	. . . . 5 ⊢ 𝑅 Se 𝐴
3		wfrlem13OLD.3	. . . . 5 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
4		wfrlem13OLD.4	. . . . 5 ⊢ 𝐶 = (𝐹 ∪ {⟨𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
5	1, 2, 3, 4	wfrlem13OLD 8123	. . . 4 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝐶 Fn (dom 𝐹 ∪ {𝑧}))
6	5	adantr 480	. . 3 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → 𝐶 Fn (dom 𝐹 ∪ {𝑧}))
7	1, 3	wfrlem10OLD 8120	. . . 4 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → Pred(𝑅, 𝐴, 𝑧) = dom 𝐹)
8		eldifi 4057	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝑧 ∈ 𝐴)
9		setlikespec 6217	. . . . . 6 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑧) ∈ V)
10	8, 2, 9	sylancl 585	. . . . 5 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → Pred(𝑅, 𝐴, 𝑧) ∈ V)
11	10	adantr 480	. . . 4 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → Pred(𝑅, 𝐴, 𝑧) ∈ V)
12	7, 11	eqeltrrd 2840	. . 3 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → dom 𝐹 ∈ V)
13		snex 5349	. . . . 5 ⊢ {𝑧} ∈ V
14		unexg 7577	. . . . 5 ⊢ ((dom 𝐹 ∈ V ∧ {𝑧} ∈ V) → (dom 𝐹 ∪ {𝑧}) ∈ V)
15	13, 14	mpan2 687	. . . 4 ⊢ (dom 𝐹 ∈ V → (dom 𝐹 ∪ {𝑧}) ∈ V)
16		fnex 7075	. . . 4 ⊢ ((𝐶 Fn (dom 𝐹 ∪ {𝑧}) ∧ (dom 𝐹 ∪ {𝑧}) ∈ V) → 𝐶 ∈ V)
17	15, 16	sylan2 592	. . 3 ⊢ ((𝐶 Fn (dom 𝐹 ∪ {𝑧}) ∧ dom 𝐹 ∈ V) → 𝐶 ∈ V)
18	6, 12, 17	syl2anc 583	. 2 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → 𝐶 ∈ V)
19	12, 13, 14	sylancl 585	. . 3 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → (dom 𝐹 ∪ {𝑧}) ∈ V)
20	3	wfrdmssOLD 8117	. . . . . . 7 ⊢ dom 𝐹 ⊆ 𝐴
21	8	snssd 4739	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → {𝑧} ⊆ 𝐴)
22		unss 4114	. . . . . . . 8 ⊢ ((dom 𝐹 ⊆ 𝐴 ∧ {𝑧} ⊆ 𝐴) ↔ (dom 𝐹 ∪ {𝑧}) ⊆ 𝐴)
23	22	biimpi 215	. . . . . . 7 ⊢ ((dom 𝐹 ⊆ 𝐴 ∧ {𝑧} ⊆ 𝐴) → (dom 𝐹 ∪ {𝑧}) ⊆ 𝐴)
24	20, 21, 23	sylancr 586	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (dom 𝐹 ∪ {𝑧}) ⊆ 𝐴)
25	24	adantr 480	. . . . 5 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → (dom 𝐹 ∪ {𝑧}) ⊆ 𝐴)
26		elun 4079	. . . . . . . 8 ⊢ (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) ↔ (𝑦 ∈ dom 𝐹 ∨ 𝑦 ∈ {𝑧}))
27		velsn 4574	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑧} ↔ 𝑦 = 𝑧)
28	27	orbi2i 909	. . . . . . . 8 ⊢ ((𝑦 ∈ dom 𝐹 ∨ 𝑦 ∈ {𝑧}) ↔ (𝑦 ∈ dom 𝐹 ∨ 𝑦 = 𝑧))
29	26, 28	bitri 274	. . . . . . 7 ⊢ (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) ↔ (𝑦 ∈ dom 𝐹 ∨ 𝑦 = 𝑧))
30	3	wfrdmclOLD 8119	. . . . . . . . . 10 ⊢ (𝑦 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑦) ⊆ dom 𝐹)
31		ssun3 4104	. . . . . . . . . 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 4102	. . . . . . . . . 10 ⊢ dom 𝐹 ⊆ (dom 𝐹 ∪ {𝑧})
35	7, 34	eqsstrdi 3971	. . . . . . . . 9 ⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → Pred(𝑅, 𝐴, 𝑧) ⊆ (dom 𝐹 ∪ {𝑧}))
36		predeq3 6195	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑧))
37	36	sseq1d 3948	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧}) ↔ Pred(𝑅, 𝐴, 𝑧) ⊆ (dom 𝐹 ∪ {𝑧})))
38	35, 37	syl5ibrcom 246	. . . . . . . 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 241	. . . . . 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 3106	. . . . 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 511	. . . 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	wfrlem14OLD 8124	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → (𝑦 ∈ (dom 𝐹 ∪ {𝑧}) → (𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
44	43	ralrimiv 3106	. . . . 5 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ∀𝑦 ∈ (dom 𝐹 ∪ {𝑧})(𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))))
45	44	adantr 480	. . . 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 1126	. . 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 6509	. . . 4 ⊢ (𝑥 = (dom 𝐹 ∪ {𝑧}) → (𝐶 Fn 𝑥 ↔ 𝐶 Fn (dom 𝐹 ∪ {𝑧})))
48		sseq1 3942	. . . . 5 ⊢ (𝑥 = (dom 𝐹 ∪ {𝑧}) → (𝑥 ⊆ 𝐴 ↔ (dom 𝐹 ∪ {𝑧}) ⊆ 𝐴))
49		sseq2 3943	. . . . . 6 ⊢ (𝑥 = (dom 𝐹 ∪ {𝑧}) → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ Pred(𝑅, 𝐴, 𝑦) ⊆ (dom 𝐹 ∪ {𝑧})))
50	49	raleqbi1dv 3331	. . . . 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 630	. . . 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 3333	. . . 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 1434	. . 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 3527	. 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 6508	. . . 4 ⊢ (𝑓 = 𝐶 → (𝑓 Fn 𝑥 ↔ 𝐶 Fn 𝑥))
56		fveq1 6755	. . . . . 6 ⊢ (𝑓 = 𝐶 → (𝑓‘𝑦) = (𝐶‘𝑦))
57		reseq1 5874	. . . . . . 7 ⊢ (𝑓 = 𝐶 → (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))
58	57	fveq2d 6760	. . . . . 6 ⊢ (𝑓 = 𝐶 → (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦))))
59	56, 58	eqeq12d 2754	. . . . 5 ⊢ (𝑓 = 𝐶 → ((𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐶‘𝑦) = (𝐺‘(𝐶 ↾ Pred(𝑅, 𝐴, 𝑦)))))
60	59	ralbidv 3120	. . . 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 1436	. . 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 1925	. 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 3605	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 395   ∨ wo 843   ∧ w3a 1085   = wceq 1539  ∃wex 1783   ∈ wcel 2108  {cab 2715  ∀wral 3063  Vcvv 3422   ∖ cdif 3880   ∪ cun 3881   ⊆ wss 3883  ∅c0 4253  {csn 4558  ⟨cop 4564   Se wse 5533   We wwe 5534  dom cdm 5580   ↾ cres 5582  Predcpred 6190   Fn wfn 6413  ‘cfv 6418  wrecscwrecs 8098

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-2nd 7805  df-frecs 8068  df-wrecs 8099

This theorem is referenced by:  wfrlem16OLD  8126