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Theorem ressffth 13828
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 13752 . . 3  |-  Rel  ( D  Func  D )
2 ressffth.d . . . . 5  |-  D  =  ( Cs  S )
3 resscat 13742 . . . . 5  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( Cs  S )  e.  Cat )
42, 3syl5eqel 2380 . . . 4  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  D  e.  Cat )
5 ressffth.i . . . . 5  |-  I  =  (idfunc `  D )
65idfucl 13771 . . . 4  |-  ( D  e.  Cat  ->  I  e.  ( D  Func  D
) )
74, 6syl 15 . . 3  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  I  e.  ( D 
Func  D ) )
8 1st2nd 6182 . . 3  |-  ( ( Rel  ( D  Func  D )  /\  I  e.  ( D  Func  D
) )  ->  I  =  <. ( 1st `  I
) ,  ( 2nd `  I ) >. )
91, 7, 8sylancr 644 . 2  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  I  =  <. ( 1st `  I ) ,  ( 2nd `  I
) >. )
10 eqidd 2297 . . . . . . . . 9  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  (  Homf 
`  D )  =  (  Homf 
`  D ) )
11 eqidd 2297 . . . . . . . . 9  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  (compf `  D )  =  (compf `  D ) )
12 eqid 2296 . . . . . . . . . . . . . 14  |-  ( Base `  C )  =  (
Base `  C )
1312ressinbas 13220 . . . . . . . . . . . . 13  |-  ( S  e.  V  ->  ( Cs  S )  =  ( Cs  ( S  i^i  ( Base `  C ) ) ) )
1413adantl 452 . . . . . . . . . . . 12  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( Cs  S )  =  ( Cs  ( S  i^i  ( Base `  C ) ) ) )
152, 14syl5eq 2340 . . . . . . . . . . 11  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  D  =  ( Cs  ( S  i^i  ( Base `  C ) ) ) )
1615fveq2d 5545 . . . . . . . . . 10  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  (  Homf 
`  D )  =  (  Homf 
`  ( Cs  ( S  i^i  ( Base `  C
) ) ) ) )
17 eqid 2296 . . . . . . . . . . . 12  |-  (  Homf  `  C )  =  (  Homf 
`  C )
18 simpl 443 . . . . . . . . . . . 12  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  C  e.  Cat )
19 inss2 3403 . . . . . . . . . . . . 13  |-  ( S  i^i  ( Base `  C
) )  C_  ( Base `  C )
2019a1i 10 . . . . . . . . . . . 12  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( S  i^i  ( Base `  C ) ) 
C_  ( Base `  C
) )
21 eqid 2296 . . . . . . . . . . . 12  |-  ( Cs  ( S  i^i  ( Base `  C ) ) )  =  ( Cs  ( S  i^i  ( Base `  C
) ) )
22 eqid 2296 . . . . . . . . . . . 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 13741 . . . . . . . . . . 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 445 . . . . . . . . . 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 2328 . . . . . . . . 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 5545 . . . . . . . . . 10  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  (compf `  D )  =  (compf `  ( Cs  ( S  i^i  ( Base `  C )
) ) ) )
2723simprd 449 . . . . . . . . . 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 2328 . . . . . . . . 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 5899 . . . . . . . . . . 11  |-  ( Cs  S )  e.  _V
302, 29eqeltri 2366 . . . . . . . . . 10  |-  D  e. 
_V
3130a1i 10 . . . . . . . . 9  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  D  e.  _V )
32 ovex 5899 . . . . . . . . . 10  |-  ( C  |`cat 
( (  Homf  `  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) ) )  e.  _V
3332a1i 10 . . . . . . . . 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 13790 . . . . . . . 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 13740 . . . . . . . . 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 13788 . . . . . . . . 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 15 . . . . . . . 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 3225 . . . . . . 7  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( D  Func  D
)  C_  ( D  Func  C ) )
3938, 7sseldd 3194 . . . . . 6  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  I  e.  ( D 
Func  C ) )
409, 39eqeltrrd 2371 . . . . 5  |-  ( ( C  e.  Cat  /\  S  e.  V )  -> 
<. ( 1st `  I
) ,  ( 2nd `  I ) >.  e.  ( D  Func  C )
)
41 df-br 4040 . . . . 5  |-  ( ( 1st `  I ) ( D  Func  C
) ( 2nd `  I
)  <->  <. ( 1st `  I
) ,  ( 2nd `  I ) >.  e.  ( D  Func  C )
)
4240, 41sylibr 203 . . . 4  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( 1st `  I
) ( D  Func  C ) ( 2nd `  I
) )
43 f1oi 5527 . . . . . 6  |-  (  _I  |`  ( x (  Hom  `  D ) y ) ) : ( x (  Hom  `  D
) y ) -1-1-onto-> ( x (  Hom  `  D
) y )
44 eqid 2296 . . . . . . . 8  |-  ( Base `  D )  =  (
Base `  D )
454adantr 451 . . . . . . . 8  |-  ( ( ( C  e.  Cat  /\  S  e.  V )  /\  ( x  e.  ( Base `  D
)  /\  y  e.  ( Base `  D )
) )  ->  D  e.  Cat )
46 eqid 2296 . . . . . . . 8  |-  (  Hom  `  D )  =  (  Hom  `  D )
47 simprl 732 . . . . . . . 8  |-  ( ( ( C  e.  Cat  /\  S  e.  V )  /\  ( x  e.  ( Base `  D
)  /\  y  e.  ( Base `  D )
) )  ->  x  e.  ( Base `  D
) )
48 simprr 733 . . . . . . . 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 13767 . . . . . . 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 2297 . . . . . . 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 2296 . . . . . . . . . 10  |-  (  Hom  `  C )  =  (  Hom  `  C )
522, 51resshom 13339 . . . . . . . . 9  |-  ( S  e.  V  ->  (  Hom  `  C )  =  (  Hom  `  D
) )
5352ad2antlr 707 . . . . . . . 8  |-  ( ( ( C  e.  Cat  /\  S  e.  V )  /\  ( x  e.  ( Base `  D
)  /\  y  e.  ( Base `  D )
) )  ->  (  Hom  `  C )  =  (  Hom  `  D
) )
545, 44, 45, 47idfu1 13770 . . . . . . . 8  |-  ( ( ( C  e.  Cat  /\  S  e.  V )  /\  ( x  e.  ( Base `  D
)  /\  y  e.  ( Base `  D )
) )  ->  (
( 1st `  I
) `  x )  =  x )
555, 44, 45, 48idfu1 13770 . . . . . . . 8  |-  ( ( ( C  e.  Cat  /\  S  e.  V )  /\  ( x  e.  ( Base `  D
)  /\  y  e.  ( Base `  D )
) )  ->  (
( 1st `  I
) `  y )  =  y )
5653, 54, 55oveq123d 5895 . . . . . . 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 5485 . . . . . 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 224 . . . . 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 2648 . . . 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 13806 . . . 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 645 . . 3  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( 1st `  I
) ( ( D Full 
C )  i^i  ( D Faith  C ) ) ( 2nd `  I ) )
62 df-br 4040 . . 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 188 . 2  |-  ( ( C  e.  Cat  /\  S  e.  V )  -> 
<. ( 1st `  I
) ,  ( 2nd `  I ) >.  e.  ( ( D Full  C )  i^i  ( D Faith  C
) ) )
649, 63eqeltrd 2370 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 358    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    i^i cin 3164    C_ wss 3165   <.cop 3656   class class class wbr 4039    _I cid 4320    X. cxp 4703    |` cres 4707   Rel wrel 4710   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   Basecbs 13164   ↾s cress 13165    Hom chom 13235   Catccat 13582    Homf chomf 13584  compfccomf 13585    |`cat cresc 13701  Subcatcsubc 13702    Func cfunc 13744  idfunccidfu 13745   Full cful 13792   Faith cfth 13793
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  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  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-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  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-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-hom 13248  df-cco 13249  df-cat 13586  df-cid 13587  df-homf 13588  df-comf 13589  df-ssc 13703  df-resc 13704  df-subc 13705  df-func 13748  df-idfu 13749  df-full 13794  df-fth 13795
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