Theorem relcoi1 6280

Description: Composition with the identity relation restricted to a relation's field. (Contributed by FL, 8-May-2011.) (Proof shortened by OpenAI, 3-Jul-2020.)

Assertion

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

Proof of Theorem relcoi1

Step	Hyp	Ref	Expression
1		coires1 6266	. 2 ⊢ (𝑅 ∘ ( I ↾ ∪ ∪ 𝑅)) = (𝑅 ↾ ∪ ∪ 𝑅)
2		relresfld 6278	. 2 ⊢ (Rel 𝑅 → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅)
3	1, 2	eqtrid 2781	1 ⊢ (Rel 𝑅 → (𝑅 ∘ ( I ↾ ∪ ∪ 𝑅)) = 𝑅)

Syntax hints:   → wi 4   = wceq 1539  ∪ cuni 4889   I cid 5559   ↾ cres 5669   ∘ ccom 5671  Rel wrel 5672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5278  ax-nul 5288  ax-pr 5414

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3421  df-v 3466  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-br 5126  df-opab 5188  df-id 5560  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679

This theorem is referenced by:  relexpsucl  15053  tsrdir  18623