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Theorem coires1 6095
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)) = (𝐴𝐵)

Proof of Theorem coires1
StepHypRef Expression
1 cocnvcnv1 6088 . . . . 5 (𝐴 ∘ I ) = (𝐴 ∘ I )
2 relcnv 5945 . . . . . 6 Rel 𝐴
3 coi1 6093 . . . . . 6 (Rel 𝐴 → (𝐴 ∘ I ) = 𝐴)
42, 3ax-mp 5 . . . . 5 (𝐴 ∘ I ) = 𝐴
51, 4eqtr3i 2847 . . . 4 (𝐴 ∘ I ) = 𝐴
65reseq1i 5827 . . 3 ((𝐴 ∘ I ) ↾ 𝐵) = (𝐴𝐵)
7 resco 6081 . . 3 ((𝐴 ∘ I ) ↾ 𝐵) = (𝐴 ∘ ( I ↾ 𝐵))
86, 7eqtr3i 2847 . 2 (𝐴𝐵) = (𝐴 ∘ ( I ↾ 𝐵))
9 rescnvcnv 6039 . 2 (𝐴𝐵) = (𝐴𝐵)
108, 9eqtr3i 2847 1 (𝐴 ∘ ( I ↾ 𝐵)) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538   I cid 5436  ccnv 5531  cres 5534  ccom 5536  Rel wrel 5537
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544
This theorem is referenced by:  relcoi1  6107  funcoeqres  6627  relexpaddg  14403  lindfres  20510  lindsmm  20515  psrass1lem  20613  kgencn2  22160  ustssco  22818  symgcom  30758  cycpmconjv  30815  cycpmconjslem1  30827  erdsze2lem2  32525  poimirlem9  35024  mzpresrename  39621  diophrw  39630  eldioph2  39633  diophren  39684  relexpiidm  40335  relexpaddss  40349  cotrclrcl  40373  funcrngcsetcALT  44562  itcoval1  45016
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