Theorem coires1 5128

Description: Composition with a restricted identity relation. (Contributed by FL, 19-Jun-2011.) (Revised by Stefan O'Rear, 7-Mar-2015.)

Assertion

Ref	Expression
coires1	⊢ (𝐴 ∘ ( I ↾ 𝐵)) = (𝐴 ↾ 𝐵)

Proof of Theorem coires1

Step	Hyp	Ref	Expression
1		cocnvcnv1 5121	. . . . 5 ⊢ (◡◡𝐴 ∘ I ) = (𝐴 ∘ I )
2		relcnv 4989	. . . . . 6 ⊢ Rel ◡◡𝐴
3		coi1 5126	. . . . . 6 ⊢ (Rel ◡◡𝐴 → (◡◡𝐴 ∘ I ) = ◡◡𝐴)
4	2, 3	ax-mp 5	. . . . 5 ⊢ (◡◡𝐴 ∘ I ) = ◡◡𝐴
5	1, 4	eqtr3i 2193	. . . 4 ⊢ (𝐴 ∘ I ) = ◡◡𝐴
6	5	reseq1i 4887	. . 3 ⊢ ((𝐴 ∘ I ) ↾ 𝐵) = (◡◡𝐴 ↾ 𝐵)
7		resco 5115	. . 3 ⊢ ((𝐴 ∘ I ) ↾ 𝐵) = (𝐴 ∘ ( I ↾ 𝐵))
8	6, 7	eqtr3i 2193	. 2 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ∘ ( I ↾ 𝐵))
9		rescnvcnv 5073	. 2 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵)
10	8, 9	eqtr3i 2193	1 ⊢ (𝐴 ∘ ( I ↾ 𝐵)) = (𝐴 ↾ 𝐵)

Syntax hints:   = wceq 1348   I cid 4273  ^◡ccnv 4610   ↾ cres 4613   ∘ ccom 4615  Rel wrel 4616

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623

This theorem is referenced by:  funcoeqres  5473