Theorem relcoi1 4849

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 4846	. . 3 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))
2		resundi 4625	. . . . 5 ⊢ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (( I ↾ dom 𝑅) ∪ ( I ↾ ran 𝑅))
3		coeq2 4494	. . . . 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 4822	. . . . . . 7 ⊢ (𝑅 ∘ (( I ↾ dom 𝑅) ∪ ( I ↾ ran 𝑅))) = ((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅)))
5		resco 4825	. . . . . . . 8 ⊢ ((𝑅 ∘ I ) ↾ dom 𝑅) = (𝑅 ∘ ( I ↾ dom 𝑅))
6		coi1 4836	. . . . . . . . 9 ⊢ (Rel 𝑅 → (𝑅 ∘ I ) = 𝑅)
7		reseq1 4606	. . . . . . . . . 10 ⊢ ((𝑅 ∘ I ) = 𝑅 → ((𝑅 ∘ I ) ↾ dom 𝑅) = (𝑅 ↾ dom 𝑅))
8		resdm 4649	. . . . . . . . . . 11 ⊢ (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
9		eqtr 2057	. . . . . . . . . . . . . 14 ⊢ ((((𝑅 ∘ I ) ↾ dom 𝑅) = (𝑅 ↾ dom 𝑅) ∧ (𝑅 ↾ dom 𝑅) = 𝑅) → ((𝑅 ∘ I ) ↾ dom 𝑅) = 𝑅)
10		eqtr 2057	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑅 ∘ ( I ↾ dom 𝑅)) = ((𝑅 ∘ I ) ↾ dom 𝑅) ∧ ((𝑅 ∘ I ) ↾ dom 𝑅) = 𝑅) → (𝑅 ∘ ( I ↾ dom 𝑅)) = 𝑅)
11		resco 4825	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∘ I ) ↾ ran 𝑅) = (𝑅 ∘ ( I ↾ ran 𝑅))
12		uneq1 3090	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅 ∘ ( I ↾ dom 𝑅)) = 𝑅 → ((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∪ (𝑅 ∘ ( I ↾ ran 𝑅))))
13		reseq1 4606	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅 ∘ I ) = 𝑅 → ((𝑅 ∘ I ) ↾ ran 𝑅) = (𝑅 ↾ ran 𝑅))
14		eqtr 2057	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 3097	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2057	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4635	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑅 ↾ ran 𝑅) ⊆ 𝑅
18		ssequn2 3116	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑅 ↾ ran 𝑅) ⊆ 𝑅 ↔ (𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅)
19	17, 18	mpbi 133	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅
20	6, 19	syl6reqr 2091	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (Rel 𝑅 → (𝑅 ∪ (𝑅 ↾ ran 𝑅)) = (𝑅 ∘ I ))
21		eqeq1 2046	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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	syl5ibr 145	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 108	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 75	. . . . . . . . . . . . . . . . . . . . . . . . . 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 108	. . . . . . . . . . . . . . . . . . . . . . . 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 2043	. . . . . . . . . . . . . . . . . . . . . . 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 75	. . . . . . . . . . . . . . . . . . . . . 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 32	. . . . . . . . . . . . . . . . . . . 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 75	. . . . . . . . . . . . . . . . . . 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 59	. . . . . . . . . . . . . . . . . 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 108	. . . . . . . . . . . . . . . 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 2043	. . . . . . . . . . . . . . 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 75	. . . . . . . . . . . . . 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 108	. . . . . . . . . . . 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 75	. . . . . . . . . . 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 32	. . . . . . . . . 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 26	. . . . . . . . 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 32	. . . . . . . 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	syl5eq 2084	. . . . . 6 ⊢ (Rel 𝑅 → (𝑅 ∘ (( I ↾ dom 𝑅) ∪ ( I ↾ ran 𝑅))) = (𝑅 ∘ I ))
46		eqeq1 2046	. . . . . 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	syl5ibr 145	. . . . 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 4607	. . . . . 6 ⊢ (∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅) → ( I ↾ ∪ ∪ 𝑅) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
50	49	coeq2d 4498	. . . . 5 ⊢ (∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅) → (𝑅 ∘ ( I ↾ ∪ ∪ 𝑅)) = (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
51	50	eqeq1d 2048	. . . 4 ⊢ (∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅) → ((𝑅 ∘ ( I ↾ ∪ ∪ 𝑅)) = (𝑅 ∘ I ) ↔ (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅 ∘ I )))
52	48, 51	syl5ibr 145	. . 3 ⊢ (∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅) → (Rel 𝑅 → (𝑅 ∘ ( I ↾ ∪ ∪ 𝑅)) = (𝑅 ∘ I )))
53	1, 52	mpcom 32	. 2 ⊢ (Rel 𝑅 → (𝑅 ∘ ( I ↾ ∪ ∪ 𝑅)) = (𝑅 ∘ I ))
54	53, 6	eqtrd 2072	1 ⊢ (Rel 𝑅 → (𝑅 ∘ ( I ↾ ∪ ∪ 𝑅)) = 𝑅)

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∪ cun 2915   ⊆ wss 2917  ∪ cuni 3580   I cid 4025  dom cdm 4345  ran crn 4346   ↾ cres 4347   ∘ ccom 4349  Rel wrel 4350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357

This theorem is referenced by: (None)