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Theorem coires1 4838
 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 4831 . . . . 5 (𝐴 ∘ I ) = (𝐴 ∘ I )
2 relcnv 4703 . . . . . 6 Rel 𝐴
3 coi1 4836 . . . . . 6 (Rel 𝐴 → (𝐴 ∘ I ) = 𝐴)
42, 3ax-mp 7 . . . . 5 (𝐴 ∘ I ) = 𝐴
51, 4eqtr3i 2062 . . . 4 (𝐴 ∘ I ) = 𝐴
65reseq1i 4608 . . 3 ((𝐴 ∘ I ) ↾ 𝐵) = (𝐴𝐵)
7 resco 4825 . . 3 ((𝐴 ∘ I ) ↾ 𝐵) = (𝐴 ∘ ( I ↾ 𝐵))
86, 7eqtr3i 2062 . 2 (𝐴𝐵) = (𝐴 ∘ ( I ↾ 𝐵))
9 rescnvcnv 4783 . 2 (𝐴𝐵) = (𝐴𝐵)
108, 9eqtr3i 2062 1 (𝐴 ∘ ( I ↾ 𝐵)) = (𝐴𝐵)
 Colors of variables: wff set class Syntax hints:   = wceq 1243   I cid 4025  ◡ccnv 4344   ↾ cres 4347   ∘ ccom 4349  Rel wrel 4350 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357 This theorem is referenced by:  funcoeqres  5157
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