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Theorem coires1 5622
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 5615 . . . . 5 (𝐴 ∘ I ) = (𝐴 ∘ I )
2 relcnv 5472 . . . . . 6 Rel 𝐴
3 coi1 5620 . . . . . 6 (Rel 𝐴 → (𝐴 ∘ I ) = 𝐴)
42, 3ax-mp 5 . . . . 5 (𝐴 ∘ I ) = 𝐴
51, 4eqtr3i 2645 . . . 4 (𝐴 ∘ I ) = 𝐴
65reseq1i 5362 . . 3 ((𝐴 ∘ I ) ↾ 𝐵) = (𝐴𝐵)
7 resco 5608 . . 3 ((𝐴 ∘ I ) ↾ 𝐵) = (𝐴 ∘ ( I ↾ 𝐵))
86, 7eqtr3i 2645 . 2 (𝐴𝐵) = (𝐴 ∘ ( I ↾ 𝐵))
9 rescnvcnv 5566 . 2 (𝐴𝐵) = (𝐴𝐵)
108, 9eqtr3i 2645 1 (𝐴 ∘ ( I ↾ 𝐵)) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480   I cid 4994  ccnv 5083  cres 5086  ccom 5088  Rel wrel 5089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096
This theorem is referenced by:  relcoi1  5633  funcoeqres  6134  relexpaddg  13743  psrass1lem  19317  lindfres  20102  lindsmm  20107  kgencn2  21300  ustssco  21958  erdsze2lem2  30947  poimirlem9  33089  mzpresrename  36832  diophrw  36841  eldioph2  36844  diophren  36896  relexpiidm  37516  relexpaddss  37530  cotrclrcl  37554  funcrngcsetcALT  41317
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