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Theorem coi2 5176
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 4853 . . 3  |-  `' ( `' A  o.  _I  )  =  ( `'  _I  o.  `' `' A
)
2 relcnv 5039 . . . . 5  |-  Rel  `' A
3 coi1 5175 . . . . 5  |-  ( Rel  `' A  ->  ( `' A  o.  _I  )  =  `' A )
42, 3ax-mp 10 . . . 4  |-  ( `' A  o.  _I  )  =  `' A
54cnveqi 4844 . . 3  |-  `' ( `' A  o.  _I  )  =  `' `' A
61, 5eqtr3i 2280 . 2  |-  ( `'  _I  o.  `' `' A )  =  `' `' A
7 dfrel2 5112 . . 3  |-  ( Rel 
A  <->  `' `' A  =  A
)
8 cnvi 5073 . . . 4  |-  `'  _I  =  _I
9 coeq2 4830 . . . . 5  |-  ( `' `' A  =  A  ->  ( `'  _I  o.  `' `' A )  =  ( `'  _I  o.  A ) )
10 coeq1 4829 . . . . 5  |-  ( `'  _I  =  _I  ->  ( `'  _I  o.  A )  =  (  _I  o.  A ) )
119, 10sylan9eq 2310 . . . 4  |-  ( ( `' `' A  =  A  /\  `'  _I  =  _I  )  ->  ( `'  _I  o.  `' `' A )  =  (  _I  o.  A ) )
128, 11mpan2 655 . . 3  |-  ( `' `' A  =  A  ->  ( `'  _I  o.  `' `' A )  =  (  _I  o.  A ) )
137, 12sylbi 189 . 2  |-  ( Rel 
A  ->  ( `'  _I  o.  `' `' A
)  =  (  _I  o.  A ) )
147biimpi 188 . 2  |-  ( Rel 
A  ->  `' `' A  =  A )
156, 13, 143eqtr3a 2314 1  |-  ( Rel 
A  ->  (  _I  o.  A )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1619    _I cid 4276   `'ccnv 4660    o. ccom 4665   Rel wrel 4666
This theorem is referenced by:  relcoi2  5187  funi  5223  fcoi2  5354  dfps2  24657
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-br 3998  df-opab 4052  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678
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