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Theorem ressffth 15795
Description: The inclusion functor from a full subcategory is a full and faithful functor, see also remark 4.4(2) in [Adamek] p. 49. (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 15719 . . 3  |-  Rel  ( D  Func  D )
2 ressffth.d . . . . 5  |-  D  =  ( Cs  S )
3 resscat 15709 . . . . 5  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( Cs  S )  e.  Cat )
42, 3syl5eqel 2512 . . . 4  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  D  e.  Cat )
5 ressffth.i . . . . 5  |-  I  =  (idfunc `  D )
65idfucl 15738 . . . 4  |-  ( D  e.  Cat  ->  I  e.  ( D  Func  D
) )
74, 6syl 17 . . 3  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  I  e.  ( D 
Func  D ) )
8 1st2nd 6844 . . 3  |-  ( ( Rel  ( D  Func  D )  /\  I  e.  ( D  Func  D
) )  ->  I  =  <. ( 1st `  I
) ,  ( 2nd `  I ) >. )
91, 7, 8sylancr 667 . 2  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  I  =  <. ( 1st `  I ) ,  ( 2nd `  I
) >. )
10 eqidd 2421 . . . . . . . . 9  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( Hom f  `  D )  =  ( Hom f  `  D ) )
11 eqidd 2421 . . . . . . . . 9  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  (compf `  D )  =  (compf `  D ) )
12 eqid 2420 . . . . . . . . . . . . . 14  |-  ( Base `  C )  =  (
Base `  C )
1312ressinbas 15145 . . . . . . . . . . . . 13  |-  ( S  e.  V  ->  ( Cs  S )  =  ( Cs  ( S  i^i  ( Base `  C ) ) ) )
1413adantl 467 . . . . . . . . . . . 12  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( Cs  S )  =  ( Cs  ( S  i^i  ( Base `  C ) ) ) )
152, 14syl5eq 2473 . . . . . . . . . . 11  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  D  =  ( Cs  ( S  i^i  ( Base `  C ) ) ) )
1615fveq2d 5876 . . . . . . . . . 10  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( Hom f  `  D )  =  ( Hom f  `  ( Cs  ( S  i^i  ( Base `  C
) ) ) ) )
17 eqid 2420 . . . . . . . . . . . 12  |-  ( Hom f  `  C )  =  ( Hom f  `  C )
18 simpl 458 . . . . . . . . . . . 12  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  C  e.  Cat )
19 inss2 3680 . . . . . . . . . . . . 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 2420 . . . . . . . . . . . 12  |-  ( Cs  ( S  i^i  ( Base `  C ) ) )  =  ( Cs  ( S  i^i  ( Base `  C
) ) )
22 eqid 2420 . . . . . . . . . . . 12  |-  ( C  |`cat 
( ( Hom f  `  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) ) )  =  ( C  |`cat 
( ( Hom f  `  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) ) )
2312, 17, 18, 20, 21, 22fullresc 15708 . . . . . . . . . . 11  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( ( Hom f  `  ( Cs  ( S  i^i  ( Base `  C ) ) ) )  =  ( Hom f  `  ( C  |`cat  ( ( Hom f  `  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) ) ) )  /\  (compf `  ( Cs  ( S  i^i  ( Base `  C ) ) ) )  =  (compf `  ( C  |`cat  ( ( Hom f  `  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) ) ) ) ) )
2423simpld 460 . . . . . . . . . 10  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( Hom f  `  ( Cs  ( S  i^i  ( Base `  C
) ) ) )  =  ( Hom f  `  ( C  |`cat 
( ( Hom f  `  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) ) ) ) )
2516, 24eqtrd 2461 . . . . . . . . 9  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( Hom f  `  D )  =  ( Hom f  `  ( C  |`cat  ( ( Hom f  `  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) ) ) ) )
2615fveq2d 5876 . . . . . . . . . 10  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  (compf `  D )  =  (compf `  ( Cs  ( S  i^i  ( Base `  C )
) ) ) )
2723simprd 464 . . . . . . . . . 10  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  (compf `  ( Cs  ( S  i^i  ( Base `  C )
) ) )  =  (compf `  ( C  |`cat  ( ( Hom f  `  C )  |`  (
( S  i^i  ( Base `  C ) )  X.  ( S  i^i  ( Base `  C )
) ) ) ) ) )
2826, 27eqtrd 2461 . . . . . . . . 9  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  (compf `  D )  =  (compf `  ( C  |`cat  ( ( Hom f  `  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) ) ) ) )
29 ovex 6324 . . . . . . . . . . 11  |-  ( Cs  S )  e.  _V
302, 29eqeltri 2504 . . . . . . . . . 10  |-  D  e. 
_V
3130a1i 11 . . . . . . . . 9  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  D  e.  _V )
32 ovex 6324 . . . . . . . . . 10  |-  ( C  |`cat 
( ( Hom f  `  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  ( ( Hom f  `  C )  |`  (
( S  i^i  ( Base `  C ) )  X.  ( S  i^i  ( Base `  C )
) ) ) )  e.  _V )
3410, 11, 25, 28, 31, 31, 31, 33funcpropd 15757 . . . . . . . 8  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( D  Func  D
)  =  ( D 
Func  ( C  |`cat  ( ( Hom f  `  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) ) ) ) )
3512, 17, 18, 20fullsubc 15707 . . . . . . . . 9  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( ( Hom f  `  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) )  e.  (Subcat `  C
) )
36 funcres2 15755 . . . . . . . . 9  |-  ( ( ( Hom f  `  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) )  e.  (Subcat `  C
)  ->  ( D  Func  ( C  |`cat  ( ( Hom f  `  C )  |`  (
( S  i^i  ( Base `  C ) )  X.  ( S  i^i  ( Base `  C )
) ) ) ) )  C_  ( D  Func  C ) )
3735, 36syl 17 . . . . . . . 8  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( D  Func  ( C  |`cat  ( ( Hom f  `  C )  |`  ( ( S  i^i  ( Base `  C )
)  X.  ( S  i^i  ( Base `  C
) ) ) ) ) )  C_  ( D  Func  C ) )
3834, 37eqsstrd 3495 . . . . . . 7  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( D  Func  D
)  C_  ( D  Func  C ) )
3938, 7sseldd 3462 . . . . . 6  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  I  e.  ( D 
Func  C ) )
409, 39eqeltrrd 2509 . . . . 5  |-  ( ( C  e.  Cat  /\  S  e.  V )  -> 
<. ( 1st `  I
) ,  ( 2nd `  I ) >.  e.  ( D  Func  C )
)
41 df-br 4418 . . . . 5  |-  ( ( 1st `  I ) ( D  Func  C
) ( 2nd `  I
)  <->  <. ( 1st `  I
) ,  ( 2nd `  I ) >.  e.  ( D  Func  C )
)
4240, 41sylibr 215 . . . 4  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( 1st `  I
) ( D  Func  C ) ( 2nd `  I
) )
43 f1oi 5857 . . . . . 6  |-  (  _I  |`  ( x ( Hom  `  D ) y ) ) : ( x ( Hom  `  D
) y ) -1-1-onto-> ( x ( Hom  `  D
) y )
44 eqid 2420 . . . . . . . 8  |-  ( Base `  D )  =  (
Base `  D )
454adantr 466 . . . . . . . 8  |-  ( ( ( C  e.  Cat  /\  S  e.  V )  /\  ( x  e.  ( Base `  D
)  /\  y  e.  ( Base `  D )
) )  ->  D  e.  Cat )
46 eqid 2420 . . . . . . . 8  |-  ( Hom  `  D )  =  ( Hom  `  D )
47 simprl 762 . . . . . . . 8  |-  ( ( ( C  e.  Cat  /\  S  e.  V )  /\  ( x  e.  ( Base `  D
)  /\  y  e.  ( Base `  D )
) )  ->  x  e.  ( Base `  D
) )
48 simprr 764 . . . . . . . 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 15734 . . . . . . 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 2421 . . . . . . 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 2420 . . . . . . . . . 10  |-  ( Hom  `  C )  =  ( Hom  `  C )
522, 51resshom 15276 . . . . . . . . 9  |-  ( S  e.  V  ->  ( Hom  `  C )  =  ( Hom  `  D
) )
5352ad2antlr 731 . . . . . . . 8  |-  ( ( ( C  e.  Cat  /\  S  e.  V )  /\  ( x  e.  ( Base `  D
)  /\  y  e.  ( Base `  D )
) )  ->  ( Hom  `  C )  =  ( Hom  `  D
) )
545, 44, 45, 47idfu1 15737 . . . . . . . 8  |-  ( ( ( C  e.  Cat  /\  S  e.  V )  /\  ( x  e.  ( Base `  D
)  /\  y  e.  ( Base `  D )
) )  ->  (
( 1st `  I
) `  x )  =  x )
555, 44, 45, 48idfu1 15737 . . . . . . . 8  |-  ( ( ( C  e.  Cat  /\  S  e.  V )  /\  ( x  e.  ( Base `  D
)  /\  y  e.  ( Base `  D )
) )  ->  (
( 1st `  I
) `  y )  =  y )
5653, 54, 55oveq123d 6317 . . . . . . 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 5819 . . . . . 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 236 . . . . 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 2844 . . . 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 15773 . . . 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 668 . . 3  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( 1st `  I
) ( ( D Full 
C )  i^i  ( D Faith  C ) ) ( 2nd `  I ) )
62 df-br 4418 . . 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 199 . 2  |-  ( ( C  e.  Cat  /\  S  e.  V )  -> 
<. ( 1st `  I
) ,  ( 2nd `  I ) >.  e.  ( ( D Full  C )  i^i  ( D Faith  C
) ) )
649, 63eqeltrd 2508 1  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  I  e.  ( ( D Full  C )  i^i  ( D Faith  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1867   A.wral 2773   _Vcvv 3078    i^i cin 3432    C_ wss 3433   <.cop 3999   class class class wbr 4417    _I cid 4755    X. cxp 4843    |` cres 4847   Rel wrel 4850   -1-1-onto->wf1o 5591   ` cfv 5592  (class class class)co 6296   1stc1st 6796   2ndc2nd 6797   Basecbs 15081   ↾s cress 15082   Hom chom 15161   Catccat 15522   Hom f chomf 15524  compfccomf 15525    |`cat cresc 15665  Subcatcsubc 15666    Func cfunc 15711  idfunccidfu 15712   Full cful 15759   Faith cfth 15760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-er 7362  df-map 7473  df-pm 7474  df-ixp 7522  df-en 7569  df-dom 7570  df-sdom 7571  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-nn 10599  df-2 10657  df-3 10658  df-4 10659  df-5 10660  df-6 10661  df-7 10662  df-8 10663  df-9 10664  df-10 10665  df-n0 10859  df-z 10927  df-dec 11041  df-ndx 15084  df-slot 15085  df-base 15086  df-sets 15087  df-ress 15088  df-hom 15174  df-cco 15175  df-cat 15526  df-cid 15527  df-homf 15528  df-comf 15529  df-ssc 15667  df-resc 15668  df-subc 15669  df-func 15715  df-idfu 15716  df-full 15761  df-fth 15762
This theorem is referenced by: (None)
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