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Theorem coi2 5524
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 5188 . . 3  |-  `' ( `' A  o.  _I  )  =  ( `'  _I  o.  `' `' A
)
2 relcnv 5374 . . . . 5  |-  Rel  `' A
3 coi1 5523 . . . . 5  |-  ( Rel  `' A  ->  ( `' A  o.  _I  )  =  `' A )
42, 3ax-mp 5 . . . 4  |-  ( `' A  o.  _I  )  =  `' A
54cnveqi 5177 . . 3  |-  `' ( `' A  o.  _I  )  =  `' `' A
61, 5eqtr3i 2498 . 2  |-  ( `'  _I  o.  `' `' A )  =  `' `' A
7 dfrel2 5457 . . 3  |-  ( Rel 
A  <->  `' `' A  =  A
)
8 cnvi 5410 . . . 4  |-  `'  _I  =  _I
9 coeq2 5161 . . . . 5  |-  ( `' `' A  =  A  ->  ( `'  _I  o.  `' `' A )  =  ( `'  _I  o.  A ) )
10 coeq1 5160 . . . . 5  |-  ( `'  _I  =  _I  ->  ( `'  _I  o.  A )  =  (  _I  o.  A ) )
119, 10sylan9eq 2528 . . . 4  |-  ( ( `' `' A  =  A  /\  `'  _I  =  _I  )  ->  ( `'  _I  o.  `' `' A )  =  (  _I  o.  A ) )
128, 11mpan2 671 . . 3  |-  ( `' `' A  =  A  ->  ( `'  _I  o.  `' `' A )  =  (  _I  o.  A ) )
137, 12sylbi 195 . 2  |-  ( Rel 
A  ->  ( `'  _I  o.  `' `' A
)  =  (  _I  o.  A ) )
147biimpi 194 . 2  |-  ( Rel 
A  ->  `' `' A  =  A )
156, 13, 143eqtr3a 2532 1  |-  ( Rel 
A  ->  (  _I  o.  A )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    _I cid 4790   `'ccnv 4998    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
This theorem is referenced by:  relcoi2  5535  funi  5618  fcoi2  5760
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