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Theorem idffth 13906
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 13835 . . 3  |-  Rel  ( C  Func  C )
2 idffth.i . . . 4  |-  I  =  (idfunc `  C )
32idfucl 13854 . . 3  |-  ( C  e.  Cat  ->  I  e.  ( C  Func  C
) )
4 1st2nd 6253 . . 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 2433 . . . . 5  |-  ( C  e.  Cat  ->  <. ( 1st `  I ) ,  ( 2nd `  I
) >.  e.  ( C 
Func  C ) )
7 df-br 4105 . . . . 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 5594 . . . . . 6  |-  (  _I  |`  ( x (  Hom  `  C ) y ) ) : ( x (  Hom  `  C
) y ) -1-1-onto-> ( x (  Hom  `  C
) y )
10 eqid 2358 . . . . . . . 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 2358 . . . . . . . 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 13850 . . . . . . 7  |-  ( ( C  e.  Cat  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  ->  ( x ( 2nd `  I ) y )  =  (  _I  |`  ( x
(  Hom  `  C ) y ) ) )
16 eqidd 2359 . . . . . . 7  |-  ( ( C  e.  Cat  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  ->  ( x (  Hom  `  C )
y )  =  ( x (  Hom  `  C
) y ) )
172, 10, 11, 13idfu1 13853 . . . . . . . 8  |-  ( ( C  e.  Cat  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  ->  ( ( 1st `  I ) `  x
)  =  x )
182, 10, 11, 14idfu1 13853 . . . . . . . 8  |-  ( ( C  e.  Cat  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  ->  ( ( 1st `  I ) `  y
)  =  y )
1917, 18oveq12d 5963 . . . . . . 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 5552 . . . . . 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 2711 . . . 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 13889 . . . 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 4105 . . 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 2432 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 1642    e. wcel 1710   A.wral 2619    i^i cin 3227   <.cop 3719   class class class wbr 4104    _I cid 4386    |` cres 4773   Rel wrel 4776   -1-1-onto->wf1o 5336   ` cfv 5337  (class class class)co 5945   1stc1st 6207   2ndc2nd 6208   Basecbs 13245    Hom chom 13316   Catccat 13665    Func cfunc 13827  idfunccidfu 13828   Full cful 13875   Faith cfth 13876
This theorem is referenced by:  rescfth  13910
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-riota 6391  df-map 6862  df-ixp 6906  df-cat 13669  df-cid 13670  df-func 13831  df-idfu 13832  df-full 13877  df-fth 13878
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