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Theorem coi2 5345
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 5015 . . 3  |-  `' ( `' A  o.  _I  )  =  ( `'  _I  o.  `' `' A
)
2 relcnv 5201 . . . . 5  |-  Rel  `' A
3 coi1 5344 . . . . 5  |-  ( Rel  `' A  ->  ( `' A  o.  _I  )  =  `' A )
42, 3ax-mp 8 . . . 4  |-  ( `' A  o.  _I  )  =  `' A
54cnveqi 5006 . . 3  |-  `' ( `' A  o.  _I  )  =  `' `' A
61, 5eqtr3i 2426 . 2  |-  ( `'  _I  o.  `' `' A )  =  `' `' A
7 dfrel2 5280 . . 3  |-  ( Rel 
A  <->  `' `' A  =  A
)
8 cnvi 5235 . . . 4  |-  `'  _I  =  _I
9 coeq2 4990 . . . . 5  |-  ( `' `' A  =  A  ->  ( `'  _I  o.  `' `' A )  =  ( `'  _I  o.  A ) )
10 coeq1 4989 . . . . 5  |-  ( `'  _I  =  _I  ->  ( `'  _I  o.  A )  =  (  _I  o.  A ) )
119, 10sylan9eq 2456 . . . 4  |-  ( ( `' `' A  =  A  /\  `'  _I  =  _I  )  ->  ( `'  _I  o.  `' `' A )  =  (  _I  o.  A ) )
128, 11mpan2 653 . . 3  |-  ( `' `' A  =  A  ->  ( `'  _I  o.  `' `' A )  =  (  _I  o.  A ) )
137, 12sylbi 188 . 2  |-  ( Rel 
A  ->  ( `'  _I  o.  `' `' A
)  =  (  _I  o.  A ) )
147biimpi 187 . 2  |-  ( Rel 
A  ->  `' `' A  =  A )
156, 13, 143eqtr3a 2460 1  |-  ( Rel 
A  ->  (  _I  o.  A )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    _I cid 4453   `'ccnv 4836    o. ccom 4841   Rel wrel 4842
This theorem is referenced by:  relcoi2  5356  funi  5442  fcoi2  5577
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 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846
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