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Theorem ressffth 14135
Description: The inclusion functor from a full subcategory is a full and faithful functor. (Contributed by Mario Carneiro, 27-Jan-2017.)
Hypotheses
Ref Expression
ressffth.d  |-  D  =  ( Cs  S )
ressffth.i  |-  I  =  (idfunc `  D )
Assertion
Ref Expression
ressffth  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  I  e.  ( ( D Full  C )  i^i  ( D Faith  C ) ) )

Proof of Theorem ressffth
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relfunc 14059 . . 3  |-  Rel  ( D  Func  D )
2 ressffth.d . . . . 5  |-  D  =  ( Cs  S )
3 resscat 14049 . . . . 5  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( Cs  S )  e.  Cat )
42, 3syl5eqel 2520 . . . 4  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  D  e.  Cat )
5 ressffth.i . . . . 5  |-  I  =  (idfunc `  D )
65idfucl 14078 . . . 4  |-  ( D  e.  Cat  ->  I  e.  ( D  Func  D
) )
74, 6syl 16 . . 3  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  I  e.  ( D 
Func  D ) )
8 1st2nd 6393 . . 3  |-  ( ( Rel  ( D  Func  D )  /\  I  e.  ( D  Func  D
) )  ->  I  =  <. ( 1st `  I
) ,  ( 2nd `  I ) >. )
91, 7, 8sylancr 645 . 2  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  I  =  <. ( 1st `  I ) ,  ( 2nd `  I
) >. )
10 eqidd 2437 . . . . . . . . 9  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  (  Homf 
`  D )  =  (  Homf 
`  D ) )
11 eqidd 2437 . . . . . . . . 9  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  (compf `  D )  =  (compf `  D ) )
12 eqid 2436 . . . . . . . . . . . . . 14  |-  ( Base `  C )  =  (
Base `  C )
1312ressinbas 13525 . . . . . . . . . . . . 13  |-  ( S  e.  V  ->  ( Cs  S )  =  ( Cs  ( S  i^i  ( Base `  C ) ) ) )
1413adantl 453 . . . . . . . . . . . 12  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( Cs  S )  =  ( Cs  ( S  i^i  ( Base `  C ) ) ) )
152, 14syl5eq 2480 . . . . . . . . . . 11  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  D  =  ( Cs  ( S  i^i  ( Base `  C ) ) ) )
1615fveq2d 5732 . . . . . . . . . 10  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  (  Homf 
`  D )  =  (  Homf 
`  ( Cs  ( S  i^i  ( Base `  C
) ) ) ) )
17 eqid 2436 . . . . . . . . . . . 12  |-  (  Homf  `  C )  =  (  Homf 
`  C )
18 simpl 444 . . . . . . . . . . . 12  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  C  e.  Cat )
19 inss2 3562 . . . . . . . . . . . . 13  |-  ( S  i^i  ( Base `  C
) )  C_  ( Base `  C )
2019a1i 11 . . . . . . . . . . . 12  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( S  i^i  ( Base `  C ) ) 
C_  ( Base `  C
) )
21 eqid 2436 . . . . . . . . . . . 12  |-  ( Cs  ( S  i^i  ( Base `  C ) ) )  =  ( Cs  ( S  i^i  ( Base `  C
) ) )
22 eqid 2436 . . . . . . . . . . . 12  |-  ( C  |`cat 
( (  Homf  `  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) ) )  =  ( C  |`cat 
( (  Homf  `  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) ) )
2312, 17, 18, 20, 21, 22fullresc 14048 . . . . . . . . . . 11  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( (  Homf  `  ( Cs  ( S  i^i  ( Base `  C ) ) ) )  =  (  Homf  `  ( C  |`cat  ( (  Homf  `  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) ) ) )  /\  (compf `  ( Cs  ( S  i^i  ( Base `  C ) ) ) )  =  (compf `  ( C  |`cat  ( (  Homf  `  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) ) ) ) ) )
2423simpld 446 . . . . . . . . . 10  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  (  Homf 
`  ( Cs  ( S  i^i  ( Base `  C
) ) ) )  =  (  Homf  `  ( C  |`cat 
( (  Homf  `  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) ) ) ) )
2516, 24eqtrd 2468 . . . . . . . . 9  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  (  Homf 
`  D )  =  (  Homf 
`  ( C  |`cat  ( (  Homf 
`  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) ) ) ) )
2615fveq2d 5732 . . . . . . . . . 10  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  (compf `  D )  =  (compf `  ( Cs  ( S  i^i  ( Base `  C )
) ) ) )
2723simprd 450 . . . . . . . . . 10  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  (compf `  ( Cs  ( S  i^i  ( Base `  C )
) ) )  =  (compf `  ( C  |`cat  ( (  Homf  `  C )  |`  (
( S  i^i  ( Base `  C ) )  X.  ( S  i^i  ( Base `  C )
) ) ) ) ) )
2826, 27eqtrd 2468 . . . . . . . . 9  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  (compf `  D )  =  (compf `  ( C  |`cat  ( (  Homf  `  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) ) ) ) )
29 ovex 6106 . . . . . . . . . . 11  |-  ( Cs  S )  e.  _V
302, 29eqeltri 2506 . . . . . . . . . 10  |-  D  e. 
_V
3130a1i 11 . . . . . . . . 9  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  D  e.  _V )
32 ovex 6106 . . . . . . . . . 10  |-  ( C  |`cat 
( (  Homf  `  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) ) )  e.  _V
3332a1i 11 . . . . . . . . 9  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( C  |`cat  ( (  Homf  `  C )  |`  (
( S  i^i  ( Base `  C ) )  X.  ( S  i^i  ( Base `  C )
) ) ) )  e.  _V )
3410, 11, 25, 28, 31, 31, 31, 33funcpropd 14097 . . . . . . . 8  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( D  Func  D
)  =  ( D 
Func  ( C  |`cat  ( (  Homf 
`  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) ) ) ) )
3512, 17, 18, 20fullsubc 14047 . . . . . . . . 9  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( (  Homf  `  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) )  e.  (Subcat `  C
) )
36 funcres2 14095 . . . . . . . . 9  |-  ( ( (  Homf 
`  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) )  e.  (Subcat `  C
)  ->  ( D  Func  ( C  |`cat  ( (  Homf  `  C )  |`  (
( S  i^i  ( Base `  C ) )  X.  ( S  i^i  ( Base `  C )
) ) ) ) )  C_  ( D  Func  C ) )
3735, 36syl 16 . . . . . . . 8  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( D  Func  ( C  |`cat  ( (  Homf  `  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) ) ) )  C_  ( D  Func  C ) )
3834, 37eqsstrd 3382 . . . . . . 7  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( D  Func  D
)  C_  ( D  Func  C ) )
3938, 7sseldd 3349 . . . . . 6  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  I  e.  ( D 
Func  C ) )
409, 39eqeltrrd 2511 . . . . 5  |-  ( ( C  e.  Cat  /\  S  e.  V )  -> 
<. ( 1st `  I
) ,  ( 2nd `  I ) >.  e.  ( D  Func  C )
)
41 df-br 4213 . . . . 5  |-  ( ( 1st `  I ) ( D  Func  C
) ( 2nd `  I
)  <->  <. ( 1st `  I
) ,  ( 2nd `  I ) >.  e.  ( D  Func  C )
)
4240, 41sylibr 204 . . . 4  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( 1st `  I
) ( D  Func  C ) ( 2nd `  I
) )
43 f1oi 5713 . . . . . 6  |-  (  _I  |`  ( x (  Hom  `  D ) y ) ) : ( x (  Hom  `  D
) y ) -1-1-onto-> ( x (  Hom  `  D
) y )
44 eqid 2436 . . . . . . . 8  |-  ( Base `  D )  =  (
Base `  D )
454adantr 452 . . . . . . . 8  |-  ( ( ( C  e.  Cat  /\  S  e.  V )  /\  ( x  e.  ( Base `  D
)  /\  y  e.  ( Base `  D )
) )  ->  D  e.  Cat )
46 eqid 2436 . . . . . . . 8  |-  (  Hom  `  D )  =  (  Hom  `  D )
47 simprl 733 . . . . . . . 8  |-  ( ( ( C  e.  Cat  /\  S  e.  V )  /\  ( x  e.  ( Base `  D
)  /\  y  e.  ( Base `  D )
) )  ->  x  e.  ( Base `  D
) )
48 simprr 734 . . . . . . . 8  |-  ( ( ( C  e.  Cat  /\  S  e.  V )  /\  ( x  e.  ( Base `  D
)  /\  y  e.  ( Base `  D )
) )  ->  y  e.  ( Base `  D
) )
495, 44, 45, 46, 47, 48idfu2nd 14074 . . . . . . 7  |-  ( ( ( C  e.  Cat  /\  S  e.  V )  /\  ( x  e.  ( Base `  D
)  /\  y  e.  ( Base `  D )
) )  ->  (
x ( 2nd `  I
) y )  =  (  _I  |`  (
x (  Hom  `  D
) y ) ) )
50 eqidd 2437 . . . . . . 7  |-  ( ( ( C  e.  Cat  /\  S  e.  V )  /\  ( x  e.  ( Base `  D
)  /\  y  e.  ( Base `  D )
) )  ->  (
x (  Hom  `  D
) y )  =  ( x (  Hom  `  D ) y ) )
51 eqid 2436 . . . . . . . . . 10  |-  (  Hom  `  C )  =  (  Hom  `  C )
522, 51resshom 13646 . . . . . . . . 9  |-  ( S  e.  V  ->  (  Hom  `  C )  =  (  Hom  `  D
) )
5352ad2antlr 708 . . . . . . . 8  |-  ( ( ( C  e.  Cat  /\  S  e.  V )  /\  ( x  e.  ( Base `  D
)  /\  y  e.  ( Base `  D )
) )  ->  (  Hom  `  C )  =  (  Hom  `  D
) )
545, 44, 45, 47idfu1 14077 . . . . . . . 8  |-  ( ( ( C  e.  Cat  /\  S  e.  V )  /\  ( x  e.  ( Base `  D
)  /\  y  e.  ( Base `  D )
) )  ->  (
( 1st `  I
) `  x )  =  x )
555, 44, 45, 48idfu1 14077 . . . . . . . 8  |-  ( ( ( C  e.  Cat  /\  S  e.  V )  /\  ( x  e.  ( Base `  D
)  /\  y  e.  ( Base `  D )
) )  ->  (
( 1st `  I
) `  y )  =  y )
5653, 54, 55oveq123d 6102 . . . . . . 7  |-  ( ( ( C  e.  Cat  /\  S  e.  V )  /\  ( x  e.  ( Base `  D
)  /\  y  e.  ( Base `  D )
) )  ->  (
( ( 1st `  I
) `  x )
(  Hom  `  C ) ( ( 1st `  I
) `  y )
)  =  ( x (  Hom  `  D
) y ) )
5749, 50, 56f1oeq123d 5671 . . . . . 6  |-  ( ( ( C  e.  Cat  /\  S  e.  V )  /\  ( x  e.  ( Base `  D
)  /\  y  e.  ( Base `  D )
) )  ->  (
( x ( 2nd `  I ) y ) : ( x (  Hom  `  D )
y ) -1-1-onto-> ( ( ( 1st `  I ) `  x
) (  Hom  `  C
) ( ( 1st `  I ) `  y
) )  <->  (  _I  |`  ( x (  Hom  `  D ) y ) ) : ( x (  Hom  `  D
) y ) -1-1-onto-> ( x (  Hom  `  D
) y ) ) )
5843, 57mpbiri 225 . . . . 5  |-  ( ( ( C  e.  Cat  /\  S  e.  V )  /\  ( x  e.  ( Base `  D
)  /\  y  e.  ( Base `  D )
) )  ->  (
x ( 2nd `  I
) y ) : ( x (  Hom  `  D ) y ) -1-1-onto-> ( ( ( 1st `  I
) `  x )
(  Hom  `  C ) ( ( 1st `  I
) `  y )
) )
5958ralrimivva 2798 . . . 4  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  A. x  e.  (
Base `  D ) A. y  e.  ( Base `  D ) ( x ( 2nd `  I
) y ) : ( x (  Hom  `  D ) y ) -1-1-onto-> ( ( ( 1st `  I
) `  x )
(  Hom  `  C ) ( ( 1st `  I
) `  y )
) )
6044, 46, 51isffth2 14113 . . . 4  |-  ( ( 1st `  I ) ( ( D Full  C
)  i^i  ( D Faith  C ) ) ( 2nd `  I )  <->  ( ( 1st `  I ) ( D  Func  C )
( 2nd `  I
)  /\  A. x  e.  ( Base `  D
) A. y  e.  ( Base `  D
) ( x ( 2nd `  I ) y ) : ( x (  Hom  `  D
) y ) -1-1-onto-> ( ( ( 1st `  I
) `  x )
(  Hom  `  C ) ( ( 1st `  I
) `  y )
) ) )
6142, 59, 60sylanbrc 646 . . 3  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( 1st `  I
) ( ( D Full 
C )  i^i  ( D Faith  C ) ) ( 2nd `  I ) )
62 df-br 4213 . . 3  |-  ( ( 1st `  I ) ( ( D Full  C
)  i^i  ( D Faith  C ) ) ( 2nd `  I )  <->  <. ( 1st `  I ) ,  ( 2nd `  I )
>.  e.  ( ( D Full 
C )  i^i  ( D Faith  C ) ) )
6361, 62sylib 189 . 2  |-  ( ( C  e.  Cat  /\  S  e.  V )  -> 
<. ( 1st `  I
) ,  ( 2nd `  I ) >.  e.  ( ( D Full  C )  i^i  ( D Faith  C
) ) )
649, 63eqeltrd 2510 1  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  I  e.  ( ( D Full  C )  i^i  ( D Faith  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   _Vcvv 2956    i^i cin 3319    C_ wss 3320   <.cop 3817   class class class wbr 4212    _I cid 4493    X. cxp 4876    |` cres 4880   Rel wrel 4883   -1-1-onto->wf1o 5453   ` cfv 5454  (class class class)co 6081   1stc1st 6347   2ndc2nd 6348   Basecbs 13469   ↾s cress 13470    Hom chom 13540   Catccat 13889    Homf chomf 13891  compfccomf 13892    |`cat cresc 14008  Subcatcsubc 14009    Func cfunc 14051  idfunccidfu 14052   Full cful 14099   Faith cfth 14100
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-map 7020  df-pm 7021  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-hom 13553  df-cco 13554  df-cat 13893  df-cid 13894  df-homf 13895  df-comf 13896  df-ssc 14010  df-resc 14011  df-subc 14012  df-func 14055  df-idfu 14056  df-full 14101  df-fth 14102
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