Theorem coires1 6233

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 6226	. . . . 5 ⊢ (◡◡𝐴 ∘ I ) = (𝐴 ∘ I )
2		relcnv 6073	. . . . . 6 ⊢ Rel ◡◡𝐴
3		coi1 6231	. . . . . 6 ⊢ (Rel ◡◡𝐴 → (◡◡𝐴 ∘ I ) = ◡◡𝐴)
4	2, 3	ax-mp 5	. . . . 5 ⊢ (◡◡𝐴 ∘ I ) = ◡◡𝐴
5	1, 4	eqtr3i 2762	. . . 4 ⊢ (𝐴 ∘ I ) = ◡◡𝐴
6	5	reseq1i 5944	. . 3 ⊢ ((𝐴 ∘ I ) ↾ 𝐵) = (◡◡𝐴 ↾ 𝐵)
7		resco 6218	. . 3 ⊢ ((𝐴 ∘ I ) ↾ 𝐵) = (𝐴 ∘ ( I ↾ 𝐵))
8	6, 7	eqtr3i 2762	. 2 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ∘ ( I ↾ 𝐵))
9		rescnvcnv 6172	. 2 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵)
10	8, 9	eqtr3i 2762	1 ⊢ (𝐴 ∘ ( I ↾ 𝐵)) = (𝐴 ↾ 𝐵)

Syntax hints:   = wceq 1542   I cid 5528  ^◡ccnv 5633   ↾ cres 5636   ∘ ccom 5638  Rel wrel 5639

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5245  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646

This theorem is referenced by:  relcoi1  6246  funcoeqres  6815  f1ofvswap  7264  relexpaddg  14990  funcrngcsetcALT  20591  lindfres  21795  lindsmm  21800  psrass1lem  21905  kgencn2  23518  ustssco  24176  symgcom  33183  cycpmconjv  33242  cycpmconjslem1  33254  erdsze2lem2  35426  poimirlem9  37909  mzpresrename  43136  diophrw  43145  eldioph2  43148  diophren  43199  relexpiidm  44089  relexpaddss  44103  cotrclrcl  44127  itcoval1  49052