Theorem dfrcl2 43665

Description: Reflexive closure of a relation as union with restricted identity relation. (Contributed by RP, 6-Jun-2020.)

Assertion

Ref	Expression
dfrcl2	⊢ r* = (𝑥 ∈ V ↦ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))

Proof of Theorem dfrcl2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rcl 43664	. 2 ⊢ r* = (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)})
2		rabab 3496	. . . . . . . 8 ⊢ {𝑧 ∈ V ∣ (𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)} = {𝑧 ∣ (𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)}
3	2	eqcomi 2745	. . . . . . 7 ⊢ {𝑧 ∣ (𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)} = {𝑧 ∈ V ∣ (𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)}
4	3	inteqi 4931	. . . . . 6 ⊢ ∩ {𝑧 ∣ (𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)} = ∩ {𝑧 ∈ V ∣ (𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)}
5	4	a1i 11	. . . . 5 ⊢ (𝑥 ∈ V → ∩ {𝑧 ∣ (𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)} = ∩ {𝑧 ∈ V ∣ (𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)})
6		vex 3468	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
7	6	dmex 7910	. . . . . . . . . 10 ⊢ dom 𝑥 ∈ V
8	6	rnex 7911	. . . . . . . . . 10 ⊢ ran 𝑥 ∈ V
9	7, 8	unex 7743	. . . . . . . . 9 ⊢ (dom 𝑥 ∪ ran 𝑥) ∈ V
10		resiexg 7913	. . . . . . . . 9 ⊢ ((dom 𝑥 ∪ ran 𝑥) ∈ V → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∈ V)
11	9, 10	ax-mp 5	. . . . . . . 8 ⊢ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∈ V
12	11, 6	unex 7743	. . . . . . 7 ⊢ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ∈ V
13	12	a1i 11	. . . . . 6 ⊢ (𝑥 ∈ V → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ∈ V)
14		ssun2 4159	. . . . . . 7 ⊢ 𝑥 ⊆ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥)
15		dmun 5895	. . . . . . . . . . . 12 ⊢ dom (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) = (dom ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ dom 𝑥)
16		dmresi 6044	. . . . . . . . . . . . 13 ⊢ dom ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥)
17	16	uneq1i 4144	. . . . . . . . . . . 12 ⊢ (dom ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ dom 𝑥) = ((dom 𝑥 ∪ ran 𝑥) ∪ dom 𝑥)
18		un23 4154	. . . . . . . . . . . . 13 ⊢ ((dom 𝑥 ∪ ran 𝑥) ∪ dom 𝑥) = ((dom 𝑥 ∪ dom 𝑥) ∪ ran 𝑥)
19		unidm 4137	. . . . . . . . . . . . . 14 ⊢ (dom 𝑥 ∪ dom 𝑥) = dom 𝑥
20	19	uneq1i 4144	. . . . . . . . . . . . 13 ⊢ ((dom 𝑥 ∪ dom 𝑥) ∪ ran 𝑥) = (dom 𝑥 ∪ ran 𝑥)
21	18, 20	eqtri 2759	. . . . . . . . . . . 12 ⊢ ((dom 𝑥 ∪ ran 𝑥) ∪ dom 𝑥) = (dom 𝑥 ∪ ran 𝑥)
22	15, 17, 21	3eqtri 2763	. . . . . . . . . . 11 ⊢ dom (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) = (dom 𝑥 ∪ ran 𝑥)
23		rnun 6139	. . . . . . . . . . . 12 ⊢ ran (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) = (ran ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ ran 𝑥)
24		rnresi 6067	. . . . . . . . . . . . 13 ⊢ ran ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥)
25	24	uneq1i 4144	. . . . . . . . . . . 12 ⊢ (ran ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ ran 𝑥) = ((dom 𝑥 ∪ ran 𝑥) ∪ ran 𝑥)
26		unass 4152	. . . . . . . . . . . . 13 ⊢ ((dom 𝑥 ∪ ran 𝑥) ∪ ran 𝑥) = (dom 𝑥 ∪ (ran 𝑥 ∪ ran 𝑥))
27		unidm 4137	. . . . . . . . . . . . . 14 ⊢ (ran 𝑥 ∪ ran 𝑥) = ran 𝑥
28	27	uneq2i 4145	. . . . . . . . . . . . 13 ⊢ (dom 𝑥 ∪ (ran 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥)
29	26, 28	eqtri 2759	. . . . . . . . . . . 12 ⊢ ((dom 𝑥 ∪ ran 𝑥) ∪ ran 𝑥) = (dom 𝑥 ∪ ran 𝑥)
30	23, 25, 29	3eqtri 2763	. . . . . . . . . . 11 ⊢ ran (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) = (dom 𝑥 ∪ ran 𝑥)
31	22, 30	uneq12i 4146	. . . . . . . . . 10 ⊢ (dom (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ∪ ran (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥)) = ((dom 𝑥 ∪ ran 𝑥) ∪ (dom 𝑥 ∪ ran 𝑥))
32		unidm 4137	. . . . . . . . . 10 ⊢ ((dom 𝑥 ∪ ran 𝑥) ∪ (dom 𝑥 ∪ ran 𝑥)) = (dom 𝑥 ∪ ran 𝑥)
33	31, 32	eqtri 2759	. . . . . . . . 9 ⊢ (dom (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ∪ ran (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥)) = (dom 𝑥 ∪ ran 𝑥)
34	33	reseq2i 5968	. . . . . . . 8 ⊢ ( I ↾ (dom (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ∪ ran (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))) = ( I ↾ (dom 𝑥 ∪ ran 𝑥))
35		ssun1 4158	. . . . . . . 8 ⊢ ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥)
36	34, 35	eqsstri 4010	. . . . . . 7 ⊢ ( I ↾ (dom (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ∪ ran (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))) ⊆ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥)
37	14, 36	pm3.2i 470	. . . . . 6 ⊢ (𝑥 ⊆ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ∧ ( I ↾ (dom (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ∪ ran (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))) ⊆ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))
38		dmeq 5888	. . . . . . . . . . 11 ⊢ (𝑧 = (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) → dom 𝑧 = dom (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))
39		rneq 5921	. . . . . . . . . . 11 ⊢ (𝑧 = (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) → ran 𝑧 = ran (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))
40	38, 39	uneq12d 4149	. . . . . . . . . 10 ⊢ (𝑧 = (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) → (dom 𝑧 ∪ ran 𝑧) = (dom (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ∪ ran (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥)))
41	40	reseq2d 5971	. . . . . . . . 9 ⊢ (𝑧 = (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) → ( I ↾ (dom 𝑧 ∪ ran 𝑧)) = ( I ↾ (dom (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ∪ ran (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))))
42		id 22	. . . . . . . . 9 ⊢ (𝑧 = (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) → 𝑧 = (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))
43	41, 42	sseq12d 3997	. . . . . . . 8 ⊢ (𝑧 = (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) → (( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧 ↔ ( I ↾ (dom (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ∪ ran (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))) ⊆ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥)))
44	43	cleq2lem 43599	. . . . . . 7 ⊢ (𝑧 = (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) → ((𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧) ↔ (𝑥 ⊆ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ∧ ( I ↾ (dom (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ∪ ran (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))) ⊆ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))))
45	44	intminss 4955	. . . . . 6 ⊢ (((( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ∈ V ∧ (𝑥 ⊆ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ∧ ( I ↾ (dom (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ∪ ran (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))) ⊆ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))) → ∩ {𝑧 ∈ V ∣ (𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)} ⊆ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))
46	13, 37, 45	sylancl 586	. . . . 5 ⊢ (𝑥 ∈ V → ∩ {𝑧 ∈ V ∣ (𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)} ⊆ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))
47	5, 46	eqsstrd 3998	. . . 4 ⊢ (𝑥 ∈ V → ∩ {𝑧 ∣ (𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)} ⊆ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))
48		dmss 5887	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ⊆ 𝑧 → dom 𝑥 ⊆ dom 𝑧)
49		rnss 5924	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ⊆ 𝑧 → ran 𝑥 ⊆ ran 𝑧)
50		unss12 4168	. . . . . . . . . . . . . . . 16 ⊢ ((dom 𝑥 ⊆ dom 𝑧 ∧ ran 𝑥 ⊆ ran 𝑧) → (dom 𝑥 ∪ ran 𝑥) ⊆ (dom 𝑧 ∪ ran 𝑧))
51	48, 49, 50	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ⊆ 𝑧 → (dom 𝑥 ∪ ran 𝑥) ⊆ (dom 𝑧 ∪ ran 𝑧))
52		dfss 3950	. . . . . . . . . . . . . . 15 ⊢ ((dom 𝑥 ∪ ran 𝑥) ⊆ (dom 𝑧 ∪ ran 𝑧) ↔ (dom 𝑥 ∪ ran 𝑥) = ((dom 𝑥 ∪ ran 𝑥) ∩ (dom 𝑧 ∪ ran 𝑧)))
53	51, 52	sylib 218	. . . . . . . . . . . . . 14 ⊢ (𝑥 ⊆ 𝑧 → (dom 𝑥 ∪ ran 𝑥) = ((dom 𝑥 ∪ ran 𝑥) ∩ (dom 𝑧 ∪ ran 𝑧)))
54		incom 4189	. . . . . . . . . . . . . 14 ⊢ ((dom 𝑥 ∪ ran 𝑥) ∩ (dom 𝑧 ∪ ran 𝑧)) = ((dom 𝑧 ∪ ran 𝑧) ∩ (dom 𝑥 ∪ ran 𝑥))
55	53, 54	eqtrdi 2787	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ 𝑧 → (dom 𝑥 ∪ ran 𝑥) = ((dom 𝑧 ∪ ran 𝑧) ∩ (dom 𝑥 ∪ ran 𝑥)))
56	55	reseq2d 5971	. . . . . . . . . . . 12 ⊢ (𝑥 ⊆ 𝑧 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ ((dom 𝑧 ∪ ran 𝑧) ∩ (dom 𝑥 ∪ ran 𝑥))))
57		resres 5984	. . . . . . . . . . . 12 ⊢ (( I ↾ (dom 𝑧 ∪ ran 𝑧)) ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ ((dom 𝑧 ∪ ran 𝑧) ∩ (dom 𝑥 ∪ ran 𝑥)))
58	56, 57	eqtr4di 2789	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝑧 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = (( I ↾ (dom 𝑧 ∪ ran 𝑧)) ↾ (dom 𝑥 ∪ ran 𝑥)))
59		resss 5993	. . . . . . . . . . . 12 ⊢ (( I ↾ (dom 𝑧 ∪ ran 𝑧)) ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ ( I ↾ (dom 𝑧 ∪ ran 𝑧))
60	59	a1i 11	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝑧 → (( I ↾ (dom 𝑧 ∪ ran 𝑧)) ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ ( I ↾ (dom 𝑧 ∪ ran 𝑧)))
61	58, 60	eqsstrd 3998	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝑧 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ ( I ↾ (dom 𝑧 ∪ ran 𝑧)))
62	61	adantr 480	. . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧) → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ ( I ↾ (dom 𝑧 ∪ ran 𝑧)))
63		simpr 484	. . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧) → ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)
64	62, 63	sstrd 3974	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧) → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧)
65		simpl 482	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧) → 𝑥 ⊆ 𝑧)
66	64, 65	unssd 4172	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧) → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ⊆ 𝑧)
67	66	ax-gen 1795	. . . . . 6 ⊢ ∀𝑧((𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧) → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ⊆ 𝑧)
68	67	a1i 11	. . . . 5 ⊢ (𝑥 ∈ V → ∀𝑧((𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧) → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ⊆ 𝑧))
69		ssintab 4946	. . . . 5 ⊢ ((( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ⊆ ∩ {𝑧 ∣ (𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)} ↔ ∀𝑧((𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧) → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ⊆ 𝑧))
70	68, 69	sylibr 234	. . . 4 ⊢ (𝑥 ∈ V → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥) ⊆ ∩ {𝑧 ∣ (𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)})
71	47, 70	eqssd 3981	. . 3 ⊢ (𝑥 ∈ V → ∩ {𝑧 ∣ (𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)} = (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))
72	71	mpteq2ia 5221	. 2 ⊢ (𝑥 ∈ V ↦ ∩ {𝑧 ∣ (𝑥 ⊆ 𝑧 ∧ ( I ↾ (dom 𝑧 ∪ ran 𝑧)) ⊆ 𝑧)}) = (𝑥 ∈ V ↦ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))
73	1, 72	eqtri 2759	1 ⊢ r* = (𝑥 ∈ V ↦ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ∪ 𝑥))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2109  {cab 2714  {crab 3420  Vcvv 3464   ∪ cun 3929   ∩ cin 3930   ⊆ wss 3931  ∩ cint 4927   ↦ cmpt 5206   I cid 5552  dom cdm 5659  ran crn 5660   ↾ cres 5661  r*crcl 43663

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 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-rcl 43664

This theorem is referenced by:  dfrcl3  43666