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Theorem coires1 5354
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 5347 . . . . 5  |-  ( `' `' A  o.  _I  )  =  ( A  o.  _I  )
2 relcnv 5209 . . . . . 6  |-  Rel  `' `' A
3 coi1 5352 . . . . . 6  |-  ( Rel  `' `' A  ->  ( `' `' A  o.  _I  )  =  `' `' A )
42, 3ax-mp 8 . . . . 5  |-  ( `' `' A  o.  _I  )  =  `' `' A
51, 4eqtr3i 2434 . . . 4  |-  ( A  o.  _I  )  =  `' `' A
65reseq1i 5109 . . 3  |-  ( ( A  o.  _I  )  |`  B )  =  ( `' `' A  |`  B )
7 resco 5341 . . 3  |-  ( ( A  o.  _I  )  |`  B )  =  ( A  o.  (  _I  |`  B ) )
86, 7eqtr3i 2434 . 2  |-  ( `' `' A  |`  B )  =  ( A  o.  (  _I  |`  B ) )
9 rescnvcnv 5299 . 2  |-  ( `' `' A  |`  B )  =  ( A  |`  B )
108, 9eqtr3i 2434 1  |-  ( A  o.  (  _I  |`  B ) )  =  ( A  |`  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    _I cid 4461   `'ccnv 4844    |` cres 4847    o. ccom 4849   Rel wrel 4850
This theorem is referenced by:  funcoeqres  5673  psrass1lem  16405  kgencn2  17550  ustssco  18205  erdsze2lem2  24851  relexpadd  25099  mzpresrename  26705  diophrw  26715  eldioph2  26718  diophren  26772  lindfres  27169  lindsmm  27174
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-opab 4235  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857
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