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Theorem idmor 25946
Description: An identity morphism. (Contributed by FL, 6-Nov-2013.)
Assertion
Ref Expression
idmor  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (
( Id SetCat `  U
) `  A )  =  <. <. A ,  A >. ,  (  _I  |`  A )
>. )

Proof of Theorem idmor
Dummy variables  a  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mptexg 5745 . . . . 5  |-  ( U  e.  Univ  ->  ( a  e.  U  |->  <. <. a ,  a >. ,  (  _I  |`  a ) >. )  e.  _V )
21adantr 451 . . . 4  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (
a  e.  U  |->  <. <. a ,  a >. ,  (  _I  |`  a
) >. )  e.  _V )
3 mpteq1 4100 . . . . 5  |-  ( u  =  U  ->  (
a  e.  u  |->  <. <. a ,  a >. ,  (  _I  |`  a
) >. )  =  ( a  e.  U  |->  <. <. a ,  a >. ,  (  _I  |`  a
) >. ) )
4 df-idcatset 25940 . . . . 5  |-  Id SetCat  =  ( u  e.  Univ  |->  ( a  e.  u  |-> 
<. <. a ,  a
>. ,  (  _I  |`  a ) >. )
)
53, 4fvmptg 5600 . . . 4  |-  ( ( U  e.  Univ  /\  (
a  e.  U  |->  <. <. a ,  a >. ,  (  _I  |`  a
) >. )  e.  _V )  ->  ( Id SetCat `  U )  =  ( a  e.  U  |->  <. <. a ,  a >. ,  (  _I  |`  a
) >. ) )
62, 5syldan 456 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ( Id SetCat `  U )  =  ( a  e.  U  |->  <. <. a ,  a
>. ,  (  _I  |`  a ) >. )
)
76fveq1d 5527 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (
( Id SetCat `  U
) `  A )  =  ( ( a  e.  U  |->  <. <. a ,  a >. ,  (  _I  |`  a ) >. ) `  A ) )
8 simpr 447 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  A  e.  U )
9 opex 4237 . . 3  |-  <. <. A ,  A >. ,  (  _I  |`  A ) >.  e.  _V
10 opeq12 3798 . . . . . 6  |-  ( ( a  =  A  /\  a  =  A )  -> 
<. a ,  a >.  =  <. A ,  A >. )
1110anidms 626 . . . . 5  |-  ( a  =  A  ->  <. a ,  a >.  =  <. A ,  A >. )
12 reseq2 4950 . . . . 5  |-  ( a  =  A  ->  (  _I  |`  a )  =  (  _I  |`  A ) )
1311, 12opeq12d 3804 . . . 4  |-  ( a  =  A  ->  <. <. a ,  a >. ,  (  _I  |`  a ) >.  =  <. <. A ,  A >. ,  (  _I  |`  A )
>. )
14 eqid 2283 . . . 4  |-  ( a  e.  U  |->  <. <. a ,  a >. ,  (  _I  |`  a ) >. )  =  ( a  e.  U  |->  <. <. a ,  a >. ,  (  _I  |`  a ) >. )
1513, 14fvmptg 5600 . . 3  |-  ( ( A  e.  U  /\  <. <. A ,  A >. ,  (  _I  |`  A )
>.  e.  _V )  -> 
( ( a  e.  U  |->  <. <. a ,  a
>. ,  (  _I  |`  a ) >. ) `  A )  =  <. <. A ,  A >. ,  (  _I  |`  A )
>. )
168, 9, 15sylancl 643 . 2  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (
( a  e.  U  |-> 
<. <. a ,  a
>. ,  (  _I  |`  a ) >. ) `  A )  =  <. <. A ,  A >. ,  (  _I  |`  A )
>. )
177, 16eqtrd 2315 1  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (
( Id SetCat `  U
) `  A )  =  <. <. A ,  A >. ,  (  _I  |`  A )
>. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643    e. cmpt 4077    _I cid 4304    |` cres 4691   ` cfv 5255   Univcgru 8412   Id SetCatcidcase 25939
This theorem is referenced by:  idmorimor  25947  domidmor  25948  codidmor  25950  grphidmor  25952  grphidmor2  25953
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-idcatset 25940
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