Theorem relcoi1 5161

Description: Composition with the identity relation restricted to a relation's field. (Contributed by FL, 8-May-2011.)

Assertion

Ref	Expression
relcoi1	⊢ (Rel 𝑅 → (𝑅 ∘ ( I ↾ ∪ ∪ 𝑅)) = 𝑅)

Proof of Theorem relcoi1

Step	Hyp	Ref	Expression
1		relfld 5158	. . 3 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))
2		resundi 4921	. . . . 5 ⊢ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (( I ↾ dom 𝑅) ∪ ( I ↾ ran 𝑅))
3		coeq2 4786	. . . . 5 ⊢ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (( I ↾ dom 𝑅) ∪ ( I ↾ ran 𝑅)) → (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅 ∘ (( I ↾ dom 𝑅) ∪ ( I ↾ ran 𝑅))))
4		coundi 5131	. . . . . . 7 ⊢ (𝑅 ∘ (( I ↾ dom 𝑅) ∪ ( I ↾ ran 𝑅))) = ((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅)))
5		resco 5134	. . . . . . . 8 ⊢ ((𝑅 ∘ I ) ↾ dom 𝑅) = (𝑅 ∘ ( I ↾ dom 𝑅))
6		coi1 5145	. . . . . . . . 9 ⊢ (Rel 𝑅 → (𝑅 ∘ I ) = 𝑅)
7		reseq1 4902	. . . . . . . . . 10 ⊢ ((𝑅 ∘ I ) = 𝑅 → ((𝑅 ∘ I ) ↾ dom 𝑅) = (𝑅 ↾ dom 𝑅))
8		resdm 4947	. . . . . . . . . . 11 ⊢ (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
9		eqtr 2195	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∘ I ) ↾ dom 𝑅) = (𝑅 ↾ dom 𝑅) ∧ (𝑅 ↾ dom 𝑅) = 𝑅) → ((𝑅 ∘ I ) ↾ dom 𝑅) = 𝑅)
10		eqtr 2195	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∘ ( I ↾ dom 𝑅)) = ((𝑅 ∘ I ) ↾ dom 𝑅) ∧ ((𝑅 ∘ I ) ↾ dom 𝑅) = 𝑅) → (𝑅 ∘ ( I ↾ dom 𝑅)) = 𝑅)
11		resco 5134	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∘ I ) ↾ ran 𝑅) = (𝑅 ∘ ( I ↾ ran 𝑅))
12		uneq1 3283	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∘ ( I ↾ dom 𝑅)) = 𝑅 → ((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∪ (𝑅 ∘ ( I ↾ ran 𝑅))))
13		reseq1 4902	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∘ I ) = 𝑅 → ((𝑅 ∘ I ) ↾ ran 𝑅) = (𝑅 ↾ ran 𝑅))
14		eqtr 2195	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑅 ∘ ( I ↾ ran 𝑅)) = ((𝑅 ∘ I ) ↾ ran 𝑅) ∧ ((𝑅 ∘ I ) ↾ ran 𝑅) = (𝑅 ↾ ran 𝑅)) → (𝑅 ∘ ( I ↾ ran 𝑅)) = (𝑅 ↾ ran 𝑅))
15	14	uneq2d 3290	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑅 ∘ ( I ↾ ran 𝑅)) = ((𝑅 ∘ I ) ↾ ran 𝑅) ∧ ((𝑅 ∘ I ) ↾ ran 𝑅) = (𝑅 ↾ ran 𝑅)) → (𝑅 ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∪ (𝑅 ↾ ran 𝑅)))
16		eqtr 2195	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) ∧ (𝑅 ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∪ (𝑅 ↾ ran 𝑅))) → ((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∪ (𝑅 ↾ ran 𝑅)))
17		resss 4932	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑅 ↾ ran 𝑅) ⊆ 𝑅
18		ssequn2 3309	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑅 ↾ ran 𝑅) ⊆ 𝑅 ↔ (𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅)
19	17, 18	mpbi 145	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅
20	19, 6	eqtr4id 2229	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (Rel 𝑅 → (𝑅 ∪ (𝑅 ↾ ran 𝑅)) = (𝑅 ∘ I ))
21		eqeq1 2184	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∪ (𝑅 ↾ ran 𝑅)) → (((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∘ I ) ↔ (𝑅 ∪ (𝑅 ↾ ran 𝑅)) = (𝑅 ∘ I )))
22	20, 21	imbitrrid 156	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∪ (𝑅 ↾ ran 𝑅)) → (Rel 𝑅 → ((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∘ I )))
23	16, 22	syl 14	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) ∧ (𝑅 ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∪ (𝑅 ↾ ran 𝑅))) → (Rel 𝑅 → ((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∘ I )))
24	23	ex 115	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) → ((𝑅 ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∪ (𝑅 ↾ ran 𝑅)) → (Rel 𝑅 → ((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∘ I ))))
25	24	com3l 81	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑅 ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∪ (𝑅 ↾ ran 𝑅)) → (Rel 𝑅 → (((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) → ((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∘ I ))))
26	15, 25	syl 14	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑅 ∘ ( I ↾ ran 𝑅)) = ((𝑅 ∘ I ) ↾ ran 𝑅) ∧ ((𝑅 ∘ I ) ↾ ran 𝑅) = (𝑅 ↾ ran 𝑅)) → (Rel 𝑅 → (((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) → ((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∘ I ))))
27	26	ex 115	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑅 ∘ ( I ↾ ran 𝑅)) = ((𝑅 ∘ I ) ↾ ran 𝑅) → (((𝑅 ∘ I ) ↾ ran 𝑅) = (𝑅 ↾ ran 𝑅) → (Rel 𝑅 → (((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) → ((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∘ I )))))
28	27	eqcoms 2180	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑅 ∘ I ) ↾ ran 𝑅) = (𝑅 ∘ ( I ↾ ran 𝑅)) → (((𝑅 ∘ I ) ↾ ran 𝑅) = (𝑅 ↾ ran 𝑅) → (Rel 𝑅 → (((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) → ((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∘ I )))))
29	28	com3l 81	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑅 ∘ I ) ↾ ran 𝑅) = (𝑅 ↾ ran 𝑅) → (Rel 𝑅 → (((𝑅 ∘ I ) ↾ ran 𝑅) = (𝑅 ∘ ( I ↾ ran 𝑅)) → (((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) → ((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∘ I )))))
30	13, 29	syl 14	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∘ I ) = 𝑅 → (Rel 𝑅 → (((𝑅 ∘ I ) ↾ ran 𝑅) = (𝑅 ∘ ( I ↾ ran 𝑅)) → (((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) → ((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∘ I )))))
31	6, 30	mpcom 36	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Rel 𝑅 → (((𝑅 ∘ I ) ↾ ran 𝑅) = (𝑅 ∘ ( I ↾ ran 𝑅)) → (((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) → ((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∘ I ))))
32	31	com3l 81	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑅 ∘ I ) ↾ ran 𝑅) = (𝑅 ∘ ( I ↾ ran 𝑅)) → (((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) → (Rel 𝑅 → ((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∘ I ))))
33	11, 12, 32	mpsyl 65	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∘ ( I ↾ dom 𝑅)) = 𝑅 → (Rel 𝑅 → ((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∘ I )))
34	10, 33	syl 14	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑅 ∘ ( I ↾ dom 𝑅)) = ((𝑅 ∘ I ) ↾ dom 𝑅) ∧ ((𝑅 ∘ I ) ↾ dom 𝑅) = 𝑅) → (Rel 𝑅 → ((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∘ I )))
35	34	ex 115	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 ∘ ( I ↾ dom 𝑅)) = ((𝑅 ∘ I ) ↾ dom 𝑅) → (((𝑅 ∘ I ) ↾ dom 𝑅) = 𝑅 → (Rel 𝑅 → ((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∘ I ))))
36	35	eqcoms 2180	. . . . . . . . . . . . . . 15 ⊢ (((𝑅 ∘ I ) ↾ dom 𝑅) = (𝑅 ∘ ( I ↾ dom 𝑅)) → (((𝑅 ∘ I ) ↾ dom 𝑅) = 𝑅 → (Rel 𝑅 → ((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∘ I ))))
37	36	com3l 81	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∘ I ) ↾ dom 𝑅) = 𝑅 → (Rel 𝑅 → (((𝑅 ∘ I ) ↾ dom 𝑅) = (𝑅 ∘ ( I ↾ dom 𝑅)) → ((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∘ I ))))
38	9, 37	syl 14	. . . . . . . . . . . . 13 ⊢ ((((𝑅 ∘ I ) ↾ dom 𝑅) = (𝑅 ↾ dom 𝑅) ∧ (𝑅 ↾ dom 𝑅) = 𝑅) → (Rel 𝑅 → (((𝑅 ∘ I ) ↾ dom 𝑅) = (𝑅 ∘ ( I ↾ dom 𝑅)) → ((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∘ I ))))
39	38	ex 115	. . . . . . . . . . . 12 ⊢ (((𝑅 ∘ I ) ↾ dom 𝑅) = (𝑅 ↾ dom 𝑅) → ((𝑅 ↾ dom 𝑅) = 𝑅 → (Rel 𝑅 → (((𝑅 ∘ I ) ↾ dom 𝑅) = (𝑅 ∘ ( I ↾ dom 𝑅)) → ((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∘ I )))))
40	39	com3l 81	. . . . . . . . . . 11 ⊢ ((𝑅 ↾ dom 𝑅) = 𝑅 → (Rel 𝑅 → (((𝑅 ∘ I ) ↾ dom 𝑅) = (𝑅 ↾ dom 𝑅) → (((𝑅 ∘ I ) ↾ dom 𝑅) = (𝑅 ∘ ( I ↾ dom 𝑅)) → ((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∘ I )))))
41	8, 40	mpcom 36	. . . . . . . . . 10 ⊢ (Rel 𝑅 → (((𝑅 ∘ I ) ↾ dom 𝑅) = (𝑅 ↾ dom 𝑅) → (((𝑅 ∘ I ) ↾ dom 𝑅) = (𝑅 ∘ ( I ↾ dom 𝑅)) → ((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∘ I ))))
42	7, 41	syl5com 29	. . . . . . . . 9 ⊢ ((𝑅 ∘ I ) = 𝑅 → (Rel 𝑅 → (((𝑅 ∘ I ) ↾ dom 𝑅) = (𝑅 ∘ ( I ↾ dom 𝑅)) → ((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∘ I ))))
43	6, 42	mpcom 36	. . . . . . . 8 ⊢ (Rel 𝑅 → (((𝑅 ∘ I ) ↾ dom 𝑅) = (𝑅 ∘ ( I ↾ dom 𝑅)) → ((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∘ I )))
44	5, 43	mpi 15	. . . . . . 7 ⊢ (Rel 𝑅 → ((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∘ I ))
45	4, 44	eqtrid 2222	. . . . . 6 ⊢ (Rel 𝑅 → (𝑅 ∘ (( I ↾ dom 𝑅) ∪ ( I ↾ ran 𝑅))) = (𝑅 ∘ I ))
46		eqeq1 2184	. . . . . 6 ⊢ ((𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅 ∘ (( I ↾ dom 𝑅) ∪ ( I ↾ ran 𝑅))) → ((𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅 ∘ I ) ↔ (𝑅 ∘ (( I ↾ dom 𝑅) ∪ ( I ↾ ran 𝑅))) = (𝑅 ∘ I )))
47	45, 46	imbitrrid 156	. . . . 5 ⊢ ((𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅 ∘ (( I ↾ dom 𝑅) ∪ ( I ↾ ran 𝑅))) → (Rel 𝑅 → (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅 ∘ I )))
48	2, 3, 47	mp2b 8	. . . 4 ⊢ (Rel 𝑅 → (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅 ∘ I ))
49		reseq2 4903	. . . . . 6 ⊢ (∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅) → ( I ↾ ∪ ∪ 𝑅) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
50	49	coeq2d 4790	. . . . 5 ⊢ (∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅) → (𝑅 ∘ ( I ↾ ∪ ∪ 𝑅)) = (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
51	50	eqeq1d 2186	. . . 4 ⊢ (∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅) → ((𝑅 ∘ ( I ↾ ∪ ∪ 𝑅)) = (𝑅 ∘ I ) ↔ (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅 ∘ I )))
52	48, 51	imbitrrid 156	. . 3 ⊢ (∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅) → (Rel 𝑅 → (𝑅 ∘ ( I ↾ ∪ ∪ 𝑅)) = (𝑅 ∘ I )))
53	1, 52	mpcom 36	. 2 ⊢ (Rel 𝑅 → (𝑅 ∘ ( I ↾ ∪ ∪ 𝑅)) = (𝑅 ∘ I ))
54	53, 6	eqtrd 2210	1 ⊢ (Rel 𝑅 → (𝑅 ∘ ( I ↾ ∪ ∪ 𝑅)) = 𝑅)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∪ cun 3128   ⊆ wss 3130  ∪ cuni 3810   I cid 4289  dom cdm 4627  ran crn 4628   ↾ cres 4629   ∘ ccom 4631  Rel wrel 4632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639

This theorem is referenced by: (None)