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Theorem coires1 6217
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 6210 . . . . 5 (𝐴 ∘ I ) = (𝐴 ∘ I )
2 relcnv 6057 . . . . . 6 Rel 𝐴
3 coi1 6215 . . . . . 6 (Rel 𝐴 → (𝐴 ∘ I ) = 𝐴)
42, 3ax-mp 5 . . . . 5 (𝐴 ∘ I ) = 𝐴
51, 4eqtr3i 2763 . . . 4 (𝐴 ∘ I ) = 𝐴
65reseq1i 5934 . . 3 ((𝐴 ∘ I ) ↾ 𝐵) = (𝐴𝐵)
7 resco 6203 . . 3 ((𝐴 ∘ I ) ↾ 𝐵) = (𝐴 ∘ ( I ↾ 𝐵))
86, 7eqtr3i 2763 . 2 (𝐴𝐵) = (𝐴 ∘ ( I ↾ 𝐵))
9 rescnvcnv 6157 . 2 (𝐴𝐵) = (𝐴𝐵)
108, 9eqtr3i 2763 1 (𝐴 ∘ ( I ↾ 𝐵)) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542   I cid 5531  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 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 5257  ax-nul 5264  ax-pr 5385
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 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  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  6231  funcoeqres  6816  f1ofvswap  7253  relexpaddg  14944  lindfres  21245  lindsmm  21250  psrass1lemOLD  21358  psrass1lem  21361  kgencn2  22924  ustssco  23582  symgcom  31983  cycpmconjv  32040  cycpmconjslem1  32052  erdsze2lem2  33855  poimirlem9  36133  mzpresrename  41116  diophrw  41125  eldioph2  41128  diophren  41179  relexpiidm  42064  relexpaddss  42078  cotrclrcl  42102  funcrngcsetcALT  46383  itcoval1  46835
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