Theorem relcoi2 6130

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 5843	. . 3 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅
2		unss 4162	. . . 4 ⊢ ((dom 𝑅 ⊆ ∪ ∪ 𝑅 ∧ ran 𝑅 ⊆ ∪ ∪ 𝑅) ↔ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅)
3		simpr 487	. . . 4 ⊢ ((dom 𝑅 ⊆ ∪ ∪ 𝑅 ∧ ran 𝑅 ⊆ ∪ ∪ 𝑅) → ran 𝑅 ⊆ ∪ ∪ 𝑅)
4	2, 3	sylbir 237	. . 3 ⊢ ((dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅 → ran 𝑅 ⊆ ∪ ∪ 𝑅)
5		cores 6104	. . 3 ⊢ (ran 𝑅 ⊆ ∪ ∪ 𝑅 → (( I ↾ ∪ ∪ 𝑅) ∘ 𝑅) = ( I ∘ 𝑅))
6	1, 4, 5	mp2b 10	. 2 ⊢ (( I ↾ ∪ ∪ 𝑅) ∘ 𝑅) = ( I ∘ 𝑅)
7		coi2 6118	. 2 ⊢ (Rel 𝑅 → ( I ∘ 𝑅) = 𝑅)
8	6, 7	syl5eq 2870	1 ⊢ (Rel 𝑅 → (( I ↾ ∪ ∪ 𝑅) ∘ 𝑅) = 𝑅)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∪ cun 3936   ⊆ wss 3938  ∪ cuni 4840   I cid 5461  dom cdm 5557  ran crn 5558   ↾ cres 5559   ∘ ccom 5561  Rel wrel 5562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569

This theorem is referenced by:  relexpsucr  14390  tsrdir  17850