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Theorem coires1 6264
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 6257 . . . . 5 (𝐴 ∘ I ) = (𝐴 ∘ I )
2 relcnv 6104 . . . . . 6 Rel 𝐴
3 coi1 6262 . . . . . 6 (Rel 𝐴 → (𝐴 ∘ I ) = 𝐴)
42, 3ax-mp 5 . . . . 5 (𝐴 ∘ I ) = 𝐴
51, 4eqtr3i 2763 . . . 4 (𝐴 ∘ I ) = 𝐴
65reseq1i 5978 . . 3 ((𝐴 ∘ I ) ↾ 𝐵) = (𝐴𝐵)
7 resco 6250 . . 3 ((𝐴 ∘ I ) ↾ 𝐵) = (𝐴 ∘ ( I ↾ 𝐵))
86, 7eqtr3i 2763 . 2 (𝐴𝐵) = (𝐴 ∘ ( I ↾ 𝐵))
9 rescnvcnv 6204 . 2 (𝐴𝐵) = (𝐴𝐵)
108, 9eqtr3i 2763 1 (𝐴 ∘ ( I ↾ 𝐵)) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542   I cid 5574  ccnv 5676  cres 5679  ccom 5681  Rel wrel 5682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689
This theorem is referenced by:  relcoi1  6278  funcoeqres  6865  f1ofvswap  7304  relexpaddg  15000  lindfres  21378  lindsmm  21383  psrass1lemOLD  21493  psrass1lem  21496  kgencn2  23061  ustssco  23719  symgcom  32244  cycpmconjv  32301  cycpmconjslem1  32313  erdsze2lem2  34195  poimirlem9  36497  mzpresrename  41488  diophrw  41497  eldioph2  41500  diophren  41551  relexpiidm  42455  relexpaddss  42469  cotrclrcl  42493  funcrngcsetcALT  46897  itcoval1  47349
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