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Theorem idffth 14135
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 14064 . . 3  |-  Rel  ( C  Func  C )
2 idffth.i . . . 4  |-  I  =  (idfunc `  C )
32idfucl 14083 . . 3  |-  ( C  e.  Cat  ->  I  e.  ( C  Func  C
) )
4 1st2nd 6396 . . 3  |-  ( ( Rel  ( C  Func  C )  /\  I  e.  ( C  Func  C
) )  ->  I  =  <. ( 1st `  I
) ,  ( 2nd `  I ) >. )
51, 3, 4sylancr 646 . 2  |-  ( C  e.  Cat  ->  I  =  <. ( 1st `  I
) ,  ( 2nd `  I ) >. )
65, 3eqeltrrd 2513 . . . . 5  |-  ( C  e.  Cat  ->  <. ( 1st `  I ) ,  ( 2nd `  I
) >.  e.  ( C 
Func  C ) )
7 df-br 4216 . . . . 5  |-  ( ( 1st `  I ) ( C  Func  C
) ( 2nd `  I
)  <->  <. ( 1st `  I
) ,  ( 2nd `  I ) >.  e.  ( C  Func  C )
)
86, 7sylibr 205 . . . 4  |-  ( C  e.  Cat  ->  ( 1st `  I ) ( C  Func  C )
( 2nd `  I
) )
9 f1oi 5716 . . . . . 6  |-  (  _I  |`  ( x (  Hom  `  C ) y ) ) : ( x (  Hom  `  C
) y ) -1-1-onto-> ( x (  Hom  `  C
) y )
10 eqid 2438 . . . . . . . 8  |-  ( Base `  C )  =  (
Base `  C )
11 simpl 445 . . . . . . . 8  |-  ( ( C  e.  Cat  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  ->  C  e.  Cat )
12 eqid 2438 . . . . . . . 8  |-  (  Hom  `  C )  =  (  Hom  `  C )
13 simprl 734 . . . . . . . 8  |-  ( ( C  e.  Cat  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  ->  x  e.  (
Base `  C )
)
14 simprr 735 . . . . . . . 8  |-  ( ( C  e.  Cat  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  ->  y  e.  (
Base `  C )
)
152, 10, 11, 12, 13, 14idfu2nd 14079 . . . . . . 7  |-  ( ( C  e.  Cat  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  ->  ( x ( 2nd `  I ) y )  =  (  _I  |`  ( x
(  Hom  `  C ) y ) ) )
16 eqidd 2439 . . . . . . 7  |-  ( ( C  e.  Cat  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  ->  ( x (  Hom  `  C )
y )  =  ( x (  Hom  `  C
) y ) )
172, 10, 11, 13idfu1 14082 . . . . . . . 8  |-  ( ( C  e.  Cat  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  ->  ( ( 1st `  I ) `  x
)  =  x )
182, 10, 11, 14idfu1 14082 . . . . . . . 8  |-  ( ( C  e.  Cat  /\  ( x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) ) )  ->  ( ( 1st `  I ) `  y
)  =  y )
1917, 18oveq12d 6102 . . . . . . 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 5674 . . . . . 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 226 . . . . 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 2800 . . . 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 14118 . . . 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 647 . . 3  |-  ( C  e.  Cat  ->  ( 1st `  I ) ( ( C Full  C )  i^i  ( C Faith  C
) ) ( 2nd `  I ) )
25 df-br 4216 . . 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 190 . 2  |-  ( C  e.  Cat  ->  <. ( 1st `  I ) ,  ( 2nd `  I
) >.  e.  ( ( C Full  C )  i^i  ( C Faith  C ) ) )
275, 26eqeltrd 2512 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 360    = wceq 1653    e. wcel 1726   A.wral 2707    i^i cin 3321   <.cop 3819   class class class wbr 4215    _I cid 4496    |` cres 4883   Rel wrel 4886   -1-1-onto->wf1o 5456   ` cfv 5457  (class class class)co 6084   1stc1st 6350   2ndc2nd 6351   Basecbs 13474    Hom chom 13545   Catccat 13894    Func cfunc 14056  idfunccidfu 14057   Full cful 14104   Faith cfth 14105
This theorem is referenced by:  rescfth  14139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-map 7023  df-ixp 7067  df-cat 13898  df-cid 13899  df-func 14060  df-idfu 14061  df-full 14106  df-fth 14107
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