Theorem relcoi2 6228

Description: Composition with the identity relation restricted to a relation's field. (Contributed by FL, 2-May-2011.)

Assertion

Ref	Expression
relcoi2	⊢ (Rel 𝑅 → (( I ↾ ∪ ∪ 𝑅) ∘ 𝑅) = 𝑅)

Proof of Theorem relcoi2

Step	Hyp	Ref	Expression
1		dmrnssfld 5916	. . 3 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅
2		unss 4119	. . . 4 ⊢ ((dom 𝑅 ⊆ ∪ ∪ 𝑅 ∧ ran 𝑅 ⊆ ∪ ∪ 𝑅) ↔ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅)
3		simpr 485	. . . 4 ⊢ ((dom 𝑅 ⊆ ∪ ∪ 𝑅 ∧ ran 𝑅 ⊆ ∪ ∪ 𝑅) → ran 𝑅 ⊆ ∪ ∪ 𝑅)
4	2, 3	sylbir 236	. . 3 ⊢ ((dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅 → ran 𝑅 ⊆ ∪ ∪ 𝑅)
5		cores 6200	. . 3 ⊢ (ran 𝑅 ⊆ ∪ ∪ 𝑅 → (( I ↾ ∪ ∪ 𝑅) ∘ 𝑅) = ( I ∘ 𝑅))
6	1, 4, 5	mp2b 10	. 2 ⊢ (( I ↾ ∪ ∪ 𝑅) ∘ 𝑅) = ( I ∘ 𝑅)
7		coi2 6215	. 2 ⊢ (Rel 𝑅 → ( I ∘ 𝑅) = 𝑅)
8	6, 7	eqtrid 2786	1 ⊢ (Rel 𝑅 → (( I ↾ ∪ ∪ 𝑅) ∘ 𝑅) = 𝑅)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∪ cun 3881   ⊆ wss 3883  ∪ cuni 4838   I cid 5512  dom cdm 5618  ran crn 5619   ↾ cres 5620   ∘ ccom 5622  Rel wrel 5623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630

This theorem is referenced by:  relexpsucr  14985  tsrdir  18561