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Theorem idcatfun 25941
Description: The identity morphims in the category Set. (Contributed by FL, 6-Nov-2013.)
Assertion
Ref Expression
idcatfun  |-  ( U  e.  Univ  ->  ( Id SetCat `
 U ) : U --> ( Morphism SetCat `  U
) )

Proof of Theorem idcatfun
Dummy variables  a  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 447 . . . 4  |-  ( ( U  e.  Univ  /\  a  e.  U )  ->  a  e.  U )
2 funi 5284 . . . . . 6  |-  Fun  _I
3 vex 2791 . . . . . 6  |-  a  e. 
_V
4 resfunexg 5737 . . . . . 6  |-  ( ( Fun  _I  /\  a  e.  _V )  ->  (  _I  |`  a )  e. 
_V )
52, 3, 4mp2an 653 . . . . 5  |-  (  _I  |`  a )  e.  _V
65a1i 10 . . . 4  |-  ( ( U  e.  Univ  /\  a  e.  U )  ->  (  _I  |`  a )  e. 
_V )
7 simpl 443 . . . 4  |-  ( ( U  e.  Univ  /\  a  e.  U )  ->  U  e.  Univ )
8 simpl1 958 . . . . 5  |-  ( ( ( a  e.  U  /\  a  e.  U  /\  (  _I  |`  a
)  e.  _V )  /\  U  e.  Univ )  ->  a  e.  U
)
9 f1oi 5511 . . . . . . 7  |-  (  _I  |`  a ) : a -1-1-onto-> a
10 f1of 5472 . . . . . . 7  |-  ( (  _I  |`  a ) : a -1-1-onto-> a  ->  (  _I  |`  a ) : a --> a )
11 elmapg 6785 . . . . . . . . . 10  |-  ( ( a  e.  _V  /\  a  e.  _V )  ->  ( (  _I  |`  a
)  e.  ( a  ^m  a )  <->  (  _I  |`  a ) : a --> a ) )
1211bicomd 192 . . . . . . . . 9  |-  ( ( a  e.  _V  /\  a  e.  _V )  ->  ( (  _I  |`  a
) : a --> a  <-> 
(  _I  |`  a
)  e.  ( a  ^m  a ) ) )
133, 3, 12mp2an 653 . . . . . . . 8  |-  ( (  _I  |`  a ) : a --> a  <->  (  _I  |`  a )  e.  ( a  ^m  a ) )
1413biimpi 186 . . . . . . 7  |-  ( (  _I  |`  a ) : a --> a  -> 
(  _I  |`  a
)  e.  ( a  ^m  a ) )
159, 10, 14mp2b 9 . . . . . 6  |-  (  _I  |`  a )  e.  ( a  ^m  a )
1615a1i 10 . . . . 5  |-  ( ( ( a  e.  U  /\  a  e.  U  /\  (  _I  |`  a
)  e.  _V )  /\  U  e.  Univ )  ->  (  _I  |`  a
)  e.  ( a  ^m  a ) )
17 prismorcset 25914 . . . . 5  |-  ( ( ( a  e.  U  /\  a  e.  U  /\  (  _I  |`  a
)  e.  _V )  /\  U  e.  Univ )  ->  ( <. <. a ,  a >. ,  (  _I  |`  a ) >.  e.  ( Morphism SetCat `  U
)  <->  ( a  e.  U  /\  a  e.  U  /\  (  _I  |`  a )  e.  ( a  ^m  a ) ) ) )
188, 8, 16, 17mpbir3and 1135 . . . 4  |-  ( ( ( a  e.  U  /\  a  e.  U  /\  (  _I  |`  a
)  e.  _V )  /\  U  e.  Univ )  ->  <. <. a ,  a
>. ,  (  _I  |`  a ) >.  e.  (
Morphism
SetCat `  U ) )
191, 1, 6, 7, 18syl31anc 1185 . . 3  |-  ( ( U  e.  Univ  /\  a  e.  U )  ->  <. <. a ,  a >. ,  (  _I  |`  a ) >.  e.  ( Morphism SetCat `  U
) )
20 eqid 2283 . . 3  |-  ( a  e.  U  |->  <. <. a ,  a >. ,  (  _I  |`  a ) >. )  =  ( a  e.  U  |->  <. <. a ,  a >. ,  (  _I  |`  a ) >. )
2119, 20fmptd 5684 . 2  |-  ( U  e.  Univ  ->  ( a  e.  U  |->  <. <. a ,  a >. ,  (  _I  |`  a ) >. ) : U --> ( Morphism SetCat `  U ) )
22 mptexg 5745 . . . 4  |-  ( U  e.  Univ  ->  ( a  e.  U  |->  <. <. a ,  a >. ,  (  _I  |`  a ) >. )  e.  _V )
23 mpteq1 4100 . . . . 5  |-  ( x  =  U  ->  (
a  e.  x  |->  <. <. a ,  a >. ,  (  _I  |`  a
) >. )  =  ( a  e.  U  |->  <. <. a ,  a >. ,  (  _I  |`  a
) >. ) )
24 df-idcatset 25940 . . . . 5  |-  Id SetCat  =  ( x  e.  Univ  |->  ( a  e.  x  |-> 
<. <. a ,  a
>. ,  (  _I  |`  a ) >. )
)
2523, 24fvmptg 5600 . . . 4  |-  ( ( U  e.  Univ  /\  (
a  e.  U  |->  <. <. a ,  a >. ,  (  _I  |`  a
) >. )  e.  _V )  ->  ( Id SetCat `  U )  =  ( a  e.  U  |->  <. <. a ,  a >. ,  (  _I  |`  a
) >. ) )
2622, 25mpdan 649 . . 3  |-  ( U  e.  Univ  ->  ( Id SetCat `
 U )  =  ( a  e.  U  |-> 
<. <. a ,  a
>. ,  (  _I  |`  a ) >. )
)
2726feq1d 5379 . 2  |-  ( U  e.  Univ  ->  ( ( Id SetCat `  U ) : U --> ( Morphism SetCat `  U
)  <->  ( a  e.  U  |->  <. <. a ,  a
>. ,  (  _I  |`  a ) >. ) : U --> ( Morphism SetCat `  U
) ) )
2821, 27mpbird 223 1  |-  ( U  e.  Univ  ->  ( Id SetCat `
 U ) : U --> ( Morphism SetCat `  U
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643    e. cmpt 4077    _I cid 4304    |` cres 4691   Fun wfun 5249   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   Univcgru 8412   Morphism SetCatccmrcase 25910   Id SetCatcidcase 25939
This theorem is referenced by:  obcatset  25942  idcatval  25943  domidcat  25945  setiscat  25979
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-13 1686  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-pow 4188  ax-pr 4214  ax-un 4512
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-pw 3627  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-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-morcatset 25911  df-idcatset 25940
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