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Theorem coires1 4888
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  o.  (  _I  |`  B ) )  =  ( A  |`  B )

Proof of Theorem coires1
StepHypRef Expression
1 cocnvcnv1 4881 . . . . 5  |-  ( `' `' A  o.  _I  )  =  ( A  o.  _I  )
2 relcnv 4753 . . . . . 6  |-  Rel  `' `' A
3 coi1 4886 . . . . . 6  |-  ( Rel  `' `' A  ->  ( `' `' A  o.  _I  )  =  `' `' A )
42, 3ax-mp 7 . . . . 5  |-  ( `' `' A  o.  _I  )  =  `' `' A
51, 4eqtr3i 2105 . . . 4  |-  ( A  o.  _I  )  =  `' `' A
65reseq1i 4656 . . 3  |-  ( ( A  o.  _I  )  |`  B )  =  ( `' `' A  |`  B )
7 resco 4875 . . 3  |-  ( ( A  o.  _I  )  |`  B )  =  ( A  o.  (  _I  |`  B ) )
86, 7eqtr3i 2105 . 2  |-  ( `' `' A  |`  B )  =  ( A  o.  (  _I  |`  B ) )
9 rescnvcnv 4833 . 2  |-  ( `' `' A  |`  B )  =  ( A  |`  B )
108, 9eqtr3i 2105 1  |-  ( A  o.  (  _I  |`  B ) )  =  ( A  |`  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1285    _I cid 4071   `'ccnv 4390    |` cres 4393    o. ccom 4395   Rel wrel 4396
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403
This theorem is referenced by:  funcoeqres  5208
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