Theorem coires1 6266

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 6259	. . . . 5 ⊢ (◡◡𝐴 ∘ I ) = (𝐴 ∘ I )
2		relcnv 6100	. . . . . 6 ⊢ Rel ◡◡𝐴
3		coi1 6264	. . . . . 6 ⊢ (Rel ◡◡𝐴 → (◡◡𝐴 ∘ I ) = ◡◡𝐴)
4	2, 3	ax-mp 5	. . . . 5 ⊢ (◡◡𝐴 ∘ I ) = ◡◡𝐴
5	1, 4	eqtr3i 2786	. . . 4 ⊢ (𝐴 ∘ I ) = ◡◡𝐴
6	5	reseq1i 5966	. . 3 ⊢ ((𝐴 ∘ I ) ↾ 𝐵) = (◡◡𝐴 ↾ 𝐵)
7		resco 6251	. . 3 ⊢ ((𝐴 ∘ I ) ↾ 𝐵) = (𝐴 ∘ ( I ↾ 𝐵))
8	6, 7	eqtr3i 2786	. 2 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ∘ ( I ↾ 𝐵))
9		rescnvcnv 6205	. 2 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵)
10	8, 9	eqtr3i 2786	1 ⊢ (𝐴 ∘ ( I ↾ 𝐵)) = (𝐴 ↾ 𝐵)

This proof depends on syntax axioms:   = wceq 1570   I cid 5545  ^◡ccnv 5650   ↾ cres 5653   ∘ ccom 5655  Rel wrel 5656

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  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 used by:  relcoi1  6281  funcoeqres  6856  f1ofvswap  7314  relexpaddg  15206  funcrngcsetcALT  20893  lindfres  22129  lindsmm  22134  psrass1lem  22241  kgencn2  23876  ustssco  24534  symgcom  33644  cycpmconjv  33703  cycpmconjslem1  33715  erdsze2lem2  35969  poimirlem9  38547  mzpresrename  43760  diophrw  43769  eldioph2  43772  diophren  43819  relexpiidm  44703  relexpaddss  44717  cotrclrcl  44741  itcoval1  49774