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Theorem coi2 5271
Description: Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)
Assertion
Ref Expression
coi2  |-  ( Rel 
A  ->  (  _I  o.  A )  =  A )

Proof of Theorem coi2
StepHypRef Expression
1 cnvco 4947 . . 3  |-  `' ( `' A  o.  _I  )  =  ( `'  _I  o.  `' `' A
)
2 relcnv 5133 . . . . 5  |-  Rel  `' A
3 coi1 5270 . . . . 5  |-  ( Rel  `' A  ->  ( `' A  o.  _I  )  =  `' A )
42, 3ax-mp 8 . . . 4  |-  ( `' A  o.  _I  )  =  `' A
54cnveqi 4938 . . 3  |-  `' ( `' A  o.  _I  )  =  `' `' A
61, 5eqtr3i 2380 . 2  |-  ( `'  _I  o.  `' `' A )  =  `' `' A
7 dfrel2 5206 . . 3  |-  ( Rel 
A  <->  `' `' A  =  A
)
8 cnvi 5167 . . . 4  |-  `'  _I  =  _I
9 coeq2 4924 . . . . 5  |-  ( `' `' A  =  A  ->  ( `'  _I  o.  `' `' A )  =  ( `'  _I  o.  A ) )
10 coeq1 4923 . . . . 5  |-  ( `'  _I  =  _I  ->  ( `'  _I  o.  A )  =  (  _I  o.  A ) )
119, 10sylan9eq 2410 . . . 4  |-  ( ( `' `' A  =  A  /\  `'  _I  =  _I  )  ->  ( `'  _I  o.  `' `' A )  =  (  _I  o.  A ) )
128, 11mpan2 652 . . 3  |-  ( `' `' A  =  A  ->  ( `'  _I  o.  `' `' A )  =  (  _I  o.  A ) )
137, 12sylbi 187 . 2  |-  ( Rel 
A  ->  ( `'  _I  o.  `' `' A
)  =  (  _I  o.  A ) )
147biimpi 186 . 2  |-  ( Rel 
A  ->  `' `' A  =  A )
156, 13, 143eqtr3a 2414 1  |-  ( Rel 
A  ->  (  _I  o.  A )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642    _I cid 4386   `'ccnv 4770    o. ccom 4775   Rel wrel 4776
This theorem is referenced by:  relcoi2  5282  funi  5366  fcoi2  5499
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4105  df-opab 4159  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780
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