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Theorem idcatfun 26044
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 5300 . . . . . 6  |-  Fun  _I
3 vex 2804 . . . . . 6  |-  a  e. 
_V
4 resfunexg 5753 . . . . . 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 5527 . . . . . . 7  |-  (  _I  |`  a ) : a -1-1-onto-> a
10 f1of 5488 . . . . . . 7  |-  ( (  _I  |`  a ) : a -1-1-onto-> a  ->  (  _I  |`  a ) : a --> a )
11 elmapg 6801 . . . . . . . . . 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 26017 . . . . 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 2296 . . 3  |-  ( a  e.  U  |->  <. <. a ,  a >. ,  (  _I  |`  a ) >. )  =  ( a  e.  U  |->  <. <. a ,  a >. ,  (  _I  |`  a ) >. )
2119, 20fmptd 5700 . 2  |-  ( U  e.  Univ  ->  ( a  e.  U  |->  <. <. a ,  a >. ,  (  _I  |`  a ) >. ) : U --> ( Morphism SetCat `  U ) )
22 mptexg 5761 . . . 4  |-  ( U  e.  Univ  ->  ( a  e.  U  |->  <. <. a ,  a >. ,  (  _I  |`  a ) >. )  e.  _V )
23 mpteq1 4116 . . . . 5  |-  ( x  =  U  ->  (
a  e.  x  |->  <. <. a ,  a >. ,  (  _I  |`  a
) >. )  =  ( a  e.  U  |->  <. <. a ,  a >. ,  (  _I  |`  a
) >. ) )
24 df-idcatset 26043 . . . . 5  |-  Id SetCat  =  ( x  e.  Univ  |->  ( a  e.  x  |-> 
<. <. a ,  a
>. ,  (  _I  |`  a ) >. )
)
2523, 24fvmptg 5616 . . . 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 5395 . 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 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656    e. cmpt 4093    _I cid 4320    |` cres 4707   Fun wfun 5265   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   Univcgru 8428   Morphism SetCatccmrcase 26013   Id SetCatcidcase 26042
This theorem is referenced by:  obcatset  26045  idcatval  26046  domidcat  26048  setiscat  26082
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-morcatset 26014  df-idcatset 26043
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