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Theorem ressffth 15168
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 15092 . . 3  |-  Rel  ( D  Func  D )
2 ressffth.d . . . . 5  |-  D  =  ( Cs  S )
3 resscat 15082 . . . . 5  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( Cs  S )  e.  Cat )
42, 3syl5eqel 2559 . . . 4  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  D  e.  Cat )
5 ressffth.i . . . . 5  |-  I  =  (idfunc `  D )
65idfucl 15111 . . . 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 6831 . . 3  |-  ( ( Rel  ( D  Func  D )  /\  I  e.  ( D  Func  D
) )  ->  I  =  <. ( 1st `  I
) ,  ( 2nd `  I ) >. )
91, 7, 8sylancr 663 . 2  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  I  =  <. ( 1st `  I ) ,  ( 2nd `  I
) >. )
10 eqidd 2468 . . . . . . . . 9  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( Hom f  `  D )  =  ( Hom f  `  D ) )
11 eqidd 2468 . . . . . . . . 9  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  (compf `  D )  =  (compf `  D ) )
12 eqid 2467 . . . . . . . . . . . . . 14  |-  ( Base `  C )  =  (
Base `  C )
1312ressinbas 14554 . . . . . . . . . . . . 13  |-  ( S  e.  V  ->  ( Cs  S )  =  ( Cs  ( S  i^i  ( Base `  C ) ) ) )
1413adantl 466 . . . . . . . . . . . 12  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( Cs  S )  =  ( Cs  ( S  i^i  ( Base `  C ) ) ) )
152, 14syl5eq 2520 . . . . . . . . . . 11  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  D  =  ( Cs  ( S  i^i  ( Base `  C ) ) ) )
1615fveq2d 5870 . . . . . . . . . 10  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( Hom f  `  D )  =  ( Hom f  `  ( Cs  ( S  i^i  ( Base `  C
) ) ) ) )
17 eqid 2467 . . . . . . . . . . . 12  |-  ( Hom f  `  C )  =  ( Hom f  `  C )
18 simpl 457 . . . . . . . . . . . 12  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  C  e.  Cat )
19 inss2 3719 . . . . . . . . . . . . 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 2467 . . . . . . . . . . . 12  |-  ( Cs  ( S  i^i  ( Base `  C ) ) )  =  ( Cs  ( S  i^i  ( Base `  C
) ) )
22 eqid 2467 . . . . . . . . . . . 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 15081 . . . . . . . . . . 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 459 . . . . . . . . . 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 2508 . . . . . . . . 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 5870 . . . . . . . . . 10  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  (compf `  D )  =  (compf `  ( Cs  ( S  i^i  ( Base `  C )
) ) ) )
2723simprd 463 . . . . . . . . . 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 2508 . . . . . . . . 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 6310 . . . . . . . . . . 11  |-  ( Cs  S )  e.  _V
302, 29eqeltri 2551 . . . . . . . . . 10  |-  D  e. 
_V
3130a1i 11 . . . . . . . . 9  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  D  e.  _V )
32 ovex 6310 . . . . . . . . . 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 15130 . . . . . . . 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 15080 . . . . . . . . 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 15128 . . . . . . . . 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 16 . . . . . . . 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 3538 . . . . . . 7  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( D  Func  D
)  C_  ( D  Func  C ) )
3938, 7sseldd 3505 . . . . . 6  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  I  e.  ( D 
Func  C ) )
409, 39eqeltrrd 2556 . . . . 5  |-  ( ( C  e.  Cat  /\  S  e.  V )  -> 
<. ( 1st `  I
) ,  ( 2nd `  I ) >.  e.  ( D  Func  C )
)
41 df-br 4448 . . . . 5  |-  ( ( 1st `  I ) ( D  Func  C
) ( 2nd `  I
)  <->  <. ( 1st `  I
) ,  ( 2nd `  I ) >.  e.  ( D  Func  C )
)
4240, 41sylibr 212 . . . 4  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( 1st `  I
) ( D  Func  C ) ( 2nd `  I
) )
43 f1oi 5851 . . . . . 6  |-  (  _I  |`  ( x ( Hom  `  D ) y ) ) : ( x ( Hom  `  D
) y ) -1-1-onto-> ( x ( Hom  `  D
) y )
44 eqid 2467 . . . . . . . 8  |-  ( Base `  D )  =  (
Base `  D )
454adantr 465 . . . . . . . 8  |-  ( ( ( C  e.  Cat  /\  S  e.  V )  /\  ( x  e.  ( Base `  D
)  /\  y  e.  ( Base `  D )
) )  ->  D  e.  Cat )
46 eqid 2467 . . . . . . . 8  |-  ( Hom  `  D )  =  ( Hom  `  D )
47 simprl 755 . . . . . . . 8  |-  ( ( ( C  e.  Cat  /\  S  e.  V )  /\  ( x  e.  ( Base `  D
)  /\  y  e.  ( Base `  D )
) )  ->  x  e.  ( Base `  D
) )
48 simprr 756 . . . . . . . 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 15107 . . . . . . 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 2468 . . . . . . 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 2467 . . . . . . . . . 10  |-  ( Hom  `  C )  =  ( Hom  `  C )
522, 51resshom 14677 . . . . . . . . 9  |-  ( S  e.  V  ->  ( Hom  `  C )  =  ( Hom  `  D
) )
5352ad2antlr 726 . . . . . . . 8  |-  ( ( ( C  e.  Cat  /\  S  e.  V )  /\  ( x  e.  ( Base `  D
)  /\  y  e.  ( Base `  D )
) )  ->  ( Hom  `  C )  =  ( Hom  `  D
) )
545, 44, 45, 47idfu1 15110 . . . . . . . 8  |-  ( ( ( C  e.  Cat  /\  S  e.  V )  /\  ( x  e.  ( Base `  D
)  /\  y  e.  ( Base `  D )
) )  ->  (
( 1st `  I
) `  x )  =  x )
555, 44, 45, 48idfu1 15110 . . . . . . . 8  |-  ( ( ( C  e.  Cat  /\  S  e.  V )  /\  ( x  e.  ( Base `  D
)  /\  y  e.  ( Base `  D )
) )  ->  (
( 1st `  I
) `  y )  =  y )
5653, 54, 55oveq123d 6306 . . . . . . 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 5813 . . . . . 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 233 . . . . 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 2885 . . . 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 15146 . . . 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 664 . . 3  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( 1st `  I
) ( ( D Full 
C )  i^i  ( D Faith  C ) ) ( 2nd `  I ) )
62 df-br 4448 . . 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 196 . 2  |-  ( ( C  e.  Cat  /\  S  e.  V )  -> 
<. ( 1st `  I
) ,  ( 2nd `  I ) >.  e.  ( ( D Full  C )  i^i  ( D Faith  C
) ) )
649, 63eqeltrd 2555 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 369    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113    i^i cin 3475    C_ wss 3476   <.cop 4033   class class class wbr 4447    _I cid 4790    X. cxp 4997    |` cres 5001   Rel wrel 5004   -1-1-onto->wf1o 5587   ` cfv 5588  (class class class)co 6285   1stc1st 6783   2ndc2nd 6784   Basecbs 14493   ↾s cress 14494   Hom chom 14569   Catccat 14922   Hom f chomf 14924  compfccomf 14925    |`cat cresc 15041  Subcatcsubc 15042    Func cfunc 15084  idfunccidfu 15085   Full cful 15132   Faith cfth 15133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7043  df-rdg 7077  df-er 7312  df-map 7423  df-pm 7424  df-ixp 7471  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10978  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-ress 14500  df-hom 14582  df-cco 14583  df-cat 14926  df-cid 14927  df-homf 14928  df-comf 14929  df-ssc 15043  df-resc 15044  df-subc 15045  df-func 15088  df-idfu 15089  df-full 15134  df-fth 15135
This theorem is referenced by: (None)
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