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Theorem idffth 13807
Description: The identity functor is a fully faithful functor. (Contributed by Mario Carneiro, 27-Jan-2017.)
Hypothesis
Ref Expression
idffth.i  |-  I  =  (idfunc `  C )
Assertion
Ref Expression
idffth  |-  ( C  e.  Cat  ->  I  e.  ( ( C Full  C
)  i^i  ( C Faith  C ) ) )

Proof of Theorem idffth
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relfunc 13736 . . 3  |-  Rel  ( C  Func  C )
2 idffth.i . . . 4  |-  I  =  (idfunc `  C )
32idfucl 13755 . . 3  |-  ( C  e.  Cat  ->  I  e.  ( C  Func  C
) )
4 1st2nd 6166 . . 3  |-  ( ( Rel  ( C  Func  C )  /\  I  e.  ( C  Func  C
) )  ->  I  =  <. ( 1st `  I
) ,  ( 2nd `  I ) >. )
51, 3, 4sylancr 644 . 2  |-  ( C  e.  Cat  ->  I  =  <. ( 1st `  I
) ,  ( 2nd `  I ) >. )
65, 3eqeltrrd 2358 . . . . 5  |-  ( C  e.  Cat  ->  <. ( 1st `  I ) ,  ( 2nd `  I
) >.  e.  ( C 
Func  C ) )
7 df-br 4024 . . . . 5  |-  ( ( 1st `  I ) ( C  Func  C
) ( 2nd `  I
)  <->  <. ( 1st `  I
) ,  ( 2nd `  I ) >.  e.  ( C  Func  C )
)
86, 7sylibr 203 . . . 4  |-  ( C  e.  Cat  ->  ( 1st `  I ) ( C  Func  C )
( 2nd `  I
) )
9 f1oi 5511 . . . . . 6  |-  (  _I  |`  ( x (  Hom  `  C ) y ) ) : ( x (  Hom  `  C
) y ) -1-1-onto-> ( x (  Hom  `  C
) y )
10 eqid 2283 . . . . . . . 8  |-  ( Base `  C )  =  (
Base `  C )
11 simpl 443 . . . . . . . 8  |-  ( ( C  e.  Cat  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  ->  C  e.  Cat )
12 eqid 2283 . . . . . . . 8  |-  (  Hom  `  C )  =  (  Hom  `  C )
13 simprl 732 . . . . . . . 8  |-  ( ( C  e.  Cat  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  ->  x  e.  (
Base `  C )
)
14 simprr 733 . . . . . . . 8  |-  ( ( C  e.  Cat  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  ->  y  e.  (
Base `  C )
)
152, 10, 11, 12, 13, 14idfu2nd 13751 . . . . . . 7  |-  ( ( C  e.  Cat  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  ->  ( x ( 2nd `  I ) y )  =  (  _I  |`  ( x
(  Hom  `  C ) y ) ) )
16 eqidd 2284 . . . . . . 7  |-  ( ( C  e.  Cat  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  ->  ( x (  Hom  `  C )
y )  =  ( x (  Hom  `  C
) y ) )
172, 10, 11, 13idfu1 13754 . . . . . . . 8  |-  ( ( C  e.  Cat  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  ->  ( ( 1st `  I ) `  x
)  =  x )
182, 10, 11, 14idfu1 13754 . . . . . . . 8  |-  ( ( C  e.  Cat  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  ->  ( ( 1st `  I ) `  y
)  =  y )
1917, 18oveq12d 5876 . . . . . . 7  |-  ( ( C  e.  Cat  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  ->  ( ( ( 1st `  I ) `
 x ) (  Hom  `  C )
( ( 1st `  I
) `  y )
)  =  ( x (  Hom  `  C
) y ) )
2015, 16, 19f1oeq123d 5469 . . . . . 6  |-  ( ( C  e.  Cat  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  ->  ( ( x ( 2nd `  I
) y ) : ( x (  Hom  `  C ) y ) -1-1-onto-> ( ( ( 1st `  I
) `  x )
(  Hom  `  C ) ( ( 1st `  I
) `  y )
)  <->  (  _I  |`  (
x (  Hom  `  C
) y ) ) : ( x (  Hom  `  C )
y ) -1-1-onto-> ( x (  Hom  `  C ) y ) ) )
219, 20mpbiri 224 . . . . 5  |-  ( ( C  e.  Cat  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  ->  ( x ( 2nd `  I ) y ) : ( x (  Hom  `  C
) y ) -1-1-onto-> ( ( ( 1st `  I
) `  x )
(  Hom  `  C ) ( ( 1st `  I
) `  y )
) )
2221ralrimivva 2635 . . . 4  |-  ( C  e.  Cat  ->  A. x  e.  ( Base `  C
) A. y  e.  ( Base `  C
) ( x ( 2nd `  I ) y ) : ( x (  Hom  `  C
) y ) -1-1-onto-> ( ( ( 1st `  I
) `  x )
(  Hom  `  C ) ( ( 1st `  I
) `  y )
) )
2310, 12, 12isffth2 13790 . . . 4  |-  ( ( 1st `  I ) ( ( C Full  C
)  i^i  ( C Faith  C ) ) ( 2nd `  I )  <->  ( ( 1st `  I ) ( C  Func  C )
( 2nd `  I
)  /\  A. x  e.  ( Base `  C
) A. y  e.  ( Base `  C
) ( x ( 2nd `  I ) y ) : ( x (  Hom  `  C
) y ) -1-1-onto-> ( ( ( 1st `  I
) `  x )
(  Hom  `  C ) ( ( 1st `  I
) `  y )
) ) )
248, 22, 23sylanbrc 645 . . 3  |-  ( C  e.  Cat  ->  ( 1st `  I ) ( ( C Full  C )  i^i  ( C Faith  C
) ) ( 2nd `  I ) )
25 df-br 4024 . . 3  |-  ( ( 1st `  I ) ( ( C Full  C
)  i^i  ( C Faith  C ) ) ( 2nd `  I )  <->  <. ( 1st `  I ) ,  ( 2nd `  I )
>.  e.  ( ( C Full 
C )  i^i  ( C Faith  C ) ) )
2624, 25sylib 188 . 2  |-  ( C  e.  Cat  ->  <. ( 1st `  I ) ,  ( 2nd `  I
) >.  e.  ( ( C Full  C )  i^i  ( C Faith  C ) ) )
275, 26eqeltrd 2357 1  |-  ( C  e.  Cat  ->  I  e.  ( ( C Full  C
)  i^i  ( C Faith  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    i^i cin 3151   <.cop 3643   class class class wbr 4023    _I cid 4304    |` cres 4691   Rel wrel 4694   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   Basecbs 13148    Hom chom 13219   Catccat 13566    Func cfunc 13728  idfunccidfu 13729   Full cful 13776   Faith cfth 13777
This theorem is referenced by:  rescfth  13811
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-rmo 2551  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-1st 6122  df-2nd 6123  df-riota 6304  df-map 6774  df-ixp 6818  df-cat 13570  df-cid 13571  df-func 13732  df-idfu 13733  df-full 13778  df-fth 13779
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