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Theorem coires1 5190
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 5183 . . . . 5  |-  ( `' `' A  o.  _I  )  =  ( A  o.  _I  )
2 relcnv 5051 . . . . . 6  |-  Rel  `' `' A
3 coi1 5188 . . . . . 6  |-  ( Rel  `' `' A  ->  ( `' `' A  o.  _I  )  =  `' `' A )
42, 3ax-mp 8 . . . . 5  |-  ( `' `' A  o.  _I  )  =  `' `' A
51, 4eqtr3i 2305 . . . 4  |-  ( A  o.  _I  )  =  `' `' A
65reseq1i 4951 . . 3  |-  ( ( A  o.  _I  )  |`  B )  =  ( `' `' A  |`  B )
7 resco 5177 . . 3  |-  ( ( A  o.  _I  )  |`  B )  =  ( A  o.  (  _I  |`  B ) )
86, 7eqtr3i 2305 . 2  |-  ( `' `' A  |`  B )  =  ( A  o.  (  _I  |`  B ) )
9 rescnvcnv 5135 . 2  |-  ( `' `' A  |`  B )  =  ( A  |`  B )
108, 9eqtr3i 2305 1  |-  ( A  o.  (  _I  |`  B ) )  =  ( A  |`  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    _I cid 4304   `'ccnv 4688    |` cres 4691    o. ccom 4693   Rel wrel 4694
This theorem is referenced by:  funcoeqres  5504  psrass1lem  16123  kgencn2  17252  erdsze2lem2  23735  relexpadd  24035  mzpresrename  26828  diophrw  26838  eldioph2  26841  diophren  26896  lindfres  27293  lindsmm  27298
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701
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