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Theorem coires1 5523
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 5516 . . . . 5  |-  ( `' `' A  o.  _I  )  =  ( A  o.  _I  )
2 relcnv 5372 . . . . . 6  |-  Rel  `' `' A
3 coi1 5521 . . . . . 6  |-  ( Rel  `' `' A  ->  ( `' `' A  o.  _I  )  =  `' `' A )
42, 3ax-mp 5 . . . . 5  |-  ( `' `' A  o.  _I  )  =  `' `' A
51, 4eqtr3i 2498 . . . 4  |-  ( A  o.  _I  )  =  `' `' A
65reseq1i 5267 . . 3  |-  ( ( A  o.  _I  )  |`  B )  =  ( `' `' A  |`  B )
7 resco 5509 . . 3  |-  ( ( A  o.  _I  )  |`  B )  =  ( A  o.  (  _I  |`  B ) )
86, 7eqtr3i 2498 . 2  |-  ( `' `' A  |`  B )  =  ( A  o.  (  _I  |`  B ) )
9 rescnvcnv 5468 . 2  |-  ( `' `' A  |`  B )  =  ( A  |`  B )
108, 9eqtr3i 2498 1  |-  ( A  o.  (  _I  |`  B ) )  =  ( A  |`  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    _I cid 4790   `'ccnv 4998    |` cres 5001    o. ccom 5003   Rel wrel 5004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011
This theorem is referenced by:  funcoeqres  5844  psrass1lem  17797  lindfres  18622  lindsmm  18627  kgencn2  19790  ustssco  20449  erdsze2lem2  28285  relexpadd  28533  mzpresrename  30285  diophrw  30294  eldioph2  30297  diophren  30349
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