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Theorem relcoi1 5040
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
StepHypRef Expression
1 relfld 5037 . . 3 (Rel 𝑅 𝑅 = (dom 𝑅 ∪ ran 𝑅))
2 resundi 4802 . . . . 5 ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = (( I ↾ dom 𝑅) ∪ ( I ↾ ran 𝑅))
3 coeq2 4667 . . . . 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 5010 . . . . . . 7 (𝑅 ∘ (( I ↾ dom 𝑅) ∪ ( I ↾ ran 𝑅))) = ((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅)))
5 resco 5013 . . . . . . . 8 ((𝑅 ∘ I ) ↾ dom 𝑅) = (𝑅 ∘ ( I ↾ dom 𝑅))
6 coi1 5024 . . . . . . . . 9 (Rel 𝑅 → (𝑅 ∘ I ) = 𝑅)
7 reseq1 4783 . . . . . . . . . 10 ((𝑅 ∘ I ) = 𝑅 → ((𝑅 ∘ I ) ↾ dom 𝑅) = (𝑅 ↾ dom 𝑅))
8 resdm 4828 . . . . . . . . . . 11 (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
9 eqtr 2135 . . . . . . . . . . . . . 14 ((((𝑅 ∘ I ) ↾ dom 𝑅) = (𝑅 ↾ dom 𝑅) ∧ (𝑅 ↾ dom 𝑅) = 𝑅) → ((𝑅 ∘ I ) ↾ dom 𝑅) = 𝑅)
10 eqtr 2135 . . . . . . . . . . . . . . . . . 18 (((𝑅 ∘ ( I ↾ dom 𝑅)) = ((𝑅 ∘ I ) ↾ dom 𝑅) ∧ ((𝑅 ∘ I ) ↾ dom 𝑅) = 𝑅) → (𝑅 ∘ ( I ↾ dom 𝑅)) = 𝑅)
11 resco 5013 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∘ I ) ↾ ran 𝑅) = (𝑅 ∘ ( I ↾ ran 𝑅))
12 uneq1 3193 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∘ ( I ↾ dom 𝑅)) = 𝑅 → ((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∪ (𝑅 ∘ ( I ↾ ran 𝑅))))
13 reseq1 4783 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑅 ∘ I ) = 𝑅 → ((𝑅 ∘ I ) ↾ ran 𝑅) = (𝑅 ↾ ran 𝑅))
14 eqtr 2135 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑅 ∘ ( I ↾ ran 𝑅)) = ((𝑅 ∘ I ) ↾ ran 𝑅) ∧ ((𝑅 ∘ I ) ↾ ran 𝑅) = (𝑅 ↾ ran 𝑅)) → (𝑅 ∘ ( I ↾ ran 𝑅)) = (𝑅 ↾ ran 𝑅))
1514uneq2d 3200 . . . . . . . . . . . . . . . . . . . . . . . . . 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 2135 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4813 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑅 ↾ ran 𝑅) ⊆ 𝑅
18 ssequn2 3219 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑅 ↾ ran 𝑅) ⊆ 𝑅 ↔ (𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅)
1917, 18mpbi 144 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑅 ∪ (𝑅 ↾ ran 𝑅)) = 𝑅
206, 19syl6reqr 2169 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (Rel 𝑅 → (𝑅 ∪ (𝑅 ↾ ran 𝑅)) = (𝑅 ∘ I ))
21 eqeq1 2124 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
2220, 21syl5ibr 155 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 )))
2316, 22syl 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 )))
2423ex 114 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ))))
2524com3l 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 ))))
2615, 25syl 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 ))))
2726ex 114 . . . . . . . . . . . . . . . . . . . . . . . 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 )))))
2827eqcoms 2120 . . . . . . . . . . . . . . . . . . . . . . 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 )))))
2928com3l 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 )))))
3013, 29syl 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 )))))
316, 30mpcom 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 ))))
3231com3l 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 ))))
3311, 12, 32mpsyl 65 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∘ ( I ↾ dom 𝑅)) = 𝑅 → (Rel 𝑅 → ((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∘ I )))
3410, 33syl 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 )))
3534ex 114 . . . . . . . . . . . . . . . 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 ))))
3635eqcoms 2120 . . . . . . . . . . . . . . 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 ))))
3736com3l 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 ))))
389, 37syl 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 ))))
3938ex 114 . . . . . . . . . . . 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 )))))
4039com3l 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 )))))
418, 40mpcom 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 ))))
427, 41syl5com 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 ))))
436, 42mpcom 36 . . . . . . . 8 (Rel 𝑅 → (((𝑅 ∘ I ) ↾ dom 𝑅) = (𝑅 ∘ ( I ↾ dom 𝑅)) → ((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∘ I )))
445, 43mpi 15 . . . . . . 7 (Rel 𝑅 → ((𝑅 ∘ ( I ↾ dom 𝑅)) ∪ (𝑅 ∘ ( I ↾ ran 𝑅))) = (𝑅 ∘ I ))
454, 44syl5eq 2162 . . . . . 6 (Rel 𝑅 → (𝑅 ∘ (( I ↾ dom 𝑅) ∪ ( I ↾ ran 𝑅))) = (𝑅 ∘ I ))
46 eqeq1 2124 . . . . . 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 )))
4745, 46syl5ibr 155 . . . . 5 ((𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅 ∘ (( I ↾ dom 𝑅) ∪ ( I ↾ ran 𝑅))) → (Rel 𝑅 → (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅 ∘ I )))
482, 3, 47mp2b 8 . . . 4 (Rel 𝑅 → (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅 ∘ I ))
49 reseq2 4784 . . . . . 6 ( 𝑅 = (dom 𝑅 ∪ ran 𝑅) → ( I ↾ 𝑅) = ( I ↾ (dom 𝑅 ∪ ran 𝑅)))
5049coeq2d 4671 . . . . 5 ( 𝑅 = (dom 𝑅 ∪ ran 𝑅) → (𝑅 ∘ ( I ↾ 𝑅)) = (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))))
5150eqeq1d 2126 . . . 4 ( 𝑅 = (dom 𝑅 ∪ ran 𝑅) → ((𝑅 ∘ ( I ↾ 𝑅)) = (𝑅 ∘ I ) ↔ (𝑅 ∘ ( I ↾ (dom 𝑅 ∪ ran 𝑅))) = (𝑅 ∘ I )))
5248, 51syl5ibr 155 . . 3 ( 𝑅 = (dom 𝑅 ∪ ran 𝑅) → (Rel 𝑅 → (𝑅 ∘ ( I ↾ 𝑅)) = (𝑅 ∘ I )))
531, 52mpcom 36 . 2 (Rel 𝑅 → (𝑅 ∘ ( I ↾ 𝑅)) = (𝑅 ∘ I ))
5453, 6eqtrd 2150 1 (Rel 𝑅 → (𝑅 ∘ ( I ↾ 𝑅)) = 𝑅)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  cun 3039  wss 3041   cuni 3706   I cid 4180  dom cdm 4509  ran crn 4510  cres 4511  ccom 4513  Rel wrel 4514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521
This theorem is referenced by: (None)
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