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Theorem coires1 5293
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 5286 . . . . 5  |-  ( `' `' A  o.  _I  )  =  ( A  o.  _I  )
2 relcnv 5154 . . . . . 6  |-  Rel  `' `' A
3 coi1 5291 . . . . . 6  |-  ( Rel  `' `' A  ->  ( `' `' A  o.  _I  )  =  `' `' A )
42, 3ax-mp 8 . . . . 5  |-  ( `' `' A  o.  _I  )  =  `' `' A
51, 4eqtr3i 2388 . . . 4  |-  ( A  o.  _I  )  =  `' `' A
65reseq1i 5054 . . 3  |-  ( ( A  o.  _I  )  |`  B )  =  ( `' `' A  |`  B )
7 resco 5280 . . 3  |-  ( ( A  o.  _I  )  |`  B )  =  ( A  o.  (  _I  |`  B ) )
86, 7eqtr3i 2388 . 2  |-  ( `' `' A  |`  B )  =  ( A  o.  (  _I  |`  B ) )
9 rescnvcnv 5238 . 2  |-  ( `' `' A  |`  B )  =  ( A  |`  B )
108, 9eqtr3i 2388 1  |-  ( A  o.  (  _I  |`  B ) )  =  ( A  |`  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1647    _I cid 4407   `'ccnv 4791    |` cres 4794    o. ccom 4796   Rel wrel 4797
This theorem is referenced by:  funcoeqres  5610  psrass1lem  16333  kgencn2  17469  ustssco  23719  erdsze2lem2  24338  relexpadd  24622  mzpresrename  26334  diophrw  26344  eldioph2  26347  diophren  26402  lindfres  26799  lindsmm  26804
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126  df-opab 4180  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804
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