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Theorem coi2 5191
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 4867 . . 3  |-  `' ( `' A  o.  _I  )  =  ( `'  _I  o.  `' `' A
)
2 relcnv 5053 . . . . 5  |-  Rel  `' A
3 coi1 5190 . . . . 5  |-  ( Rel  `' A  ->  ( `' A  o.  _I  )  =  `' A )
42, 3ax-mp 8 . . . 4  |-  ( `' A  o.  _I  )  =  `' A
54cnveqi 4858 . . 3  |-  `' ( `' A  o.  _I  )  =  `' `' A
61, 5eqtr3i 2307 . 2  |-  ( `'  _I  o.  `' `' A )  =  `' `' A
7 dfrel2 5126 . . 3  |-  ( Rel 
A  <->  `' `' A  =  A
)
8 cnvi 5087 . . . 4  |-  `'  _I  =  _I
9 coeq2 4844 . . . . 5  |-  ( `' `' A  =  A  ->  ( `'  _I  o.  `' `' A )  =  ( `'  _I  o.  A ) )
10 coeq1 4843 . . . . 5  |-  ( `'  _I  =  _I  ->  ( `'  _I  o.  A )  =  (  _I  o.  A ) )
119, 10sylan9eq 2337 . . . 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 2341 1  |-  ( Rel 
A  ->  (  _I  o.  A )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1625    _I cid 4306   `'ccnv 4690    o. ccom 4695   Rel wrel 4696
This theorem is referenced by:  relcoi2  5202  funi  5286  fcoi2  5418  dfps2  25300
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-br 4026  df-opab 4080  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700
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