Theorem coires1 6255

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 6248	. . . . 5 ⊢ (◡◡𝐴 ∘ I ) = (𝐴 ∘ I )
2		relcnv 6096	. . . . . 6 ⊢ Rel ◡◡𝐴
3		coi1 6253	. . . . . 6 ⊢ (Rel ◡◡𝐴 → (◡◡𝐴 ∘ I ) = ◡◡𝐴)
4	2, 3	ax-mp 5	. . . . 5 ⊢ (◡◡𝐴 ∘ I ) = ◡◡𝐴
5	1, 4	eqtr3i 2790	. . . 4 ⊢ (𝐴 ∘ I ) = ◡◡𝐴
6	5	reseq1i 5964	. . 3 ⊢ ((𝐴 ∘ I ) ↾ 𝐵) = (◡◡𝐴 ↾ 𝐵)
7		resco 6240	. . 3 ⊢ ((𝐴 ∘ I ) ↾ 𝐵) = (𝐴 ∘ ( I ↾ 𝐵))
8	6, 7	eqtr3i 2790	. 2 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ∘ ( I ↾ 𝐵))
9		rescnvcnv 6194	. 2 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵)
10	8, 9	eqtr3i 2790	1 ⊢ (𝐴 ∘ ( I ↾ 𝐵)) = (𝐴 ↾ 𝐵)

Syntax hints:   = wceq 1563   I cid 5545  ^◡ccnv 5650   ↾ cres 5653   ∘ ccom 5655  Rel wrel 5656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5250  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663

This theorem is referenced by:  relcoi1  6268  funcoeqres  6842  f1ofvswap  7294  relexpaddg  15078  funcrngcsetcALT  20714  lindfres  21930  lindsmm  21935  psrass1lem  22040  kgencn2  23671  ustssco  24329  symgcom  33311  cycpmconjv  33370  cycpmconjslem1  33382  erdsze2lem2  35562  poimirlem9  38135  mzpresrename  43338  diophrw  43347  eldioph2  43350  diophren  43397  relexpiidm  44287  relexpaddss  44301  cotrclrcl  44325  itcoval1  49295