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Theorem ressffth 13812
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 13736 . . 3  |-  Rel  ( D  Func  D )
2 ressffth.d . . . . 5  |-  D  =  ( Cs  S )
3 resscat 13726 . . . . 5  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( Cs  S )  e.  Cat )
42, 3syl5eqel 2367 . . . 4  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  D  e.  Cat )
5 ressffth.i . . . . 5  |-  I  =  (idfunc `  D )
65idfucl 13755 . . . 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 6166 . . 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 2284 . . . . . . . . 9  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  (  Homf 
`  D )  =  (  Homf 
`  D ) )
11 eqidd 2284 . . . . . . . . 9  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  (compf `  D )  =  (compf `  D ) )
12 eqid 2283 . . . . . . . . . . . . . 14  |-  ( Base `  C )  =  (
Base `  C )
1312ressinbas 13204 . . . . . . . . . . . . 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 2327 . . . . . . . . . . 11  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  D  =  ( Cs  ( S  i^i  ( Base `  C ) ) ) )
1615fveq2d 5529 . . . . . . . . . 10  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  (  Homf 
`  D )  =  (  Homf 
`  ( Cs  ( S  i^i  ( Base `  C
) ) ) ) )
17 eqid 2283 . . . . . . . . . . . 12  |-  (  Homf  `  C )  =  (  Homf 
`  C )
18 simpl 443 . . . . . . . . . . . 12  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  C  e.  Cat )
19 inss2 3390 . . . . . . . . . . . . 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 2283 . . . . . . . . . . . 12  |-  ( Cs  ( S  i^i  ( Base `  C ) ) )  =  ( Cs  ( S  i^i  ( Base `  C
) ) )
22 eqid 2283 . . . . . . . . . . . 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 13725 . . . . . . . . . . 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 2315 . . . . . . . . 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 5529 . . . . . . . . . 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 2315 . . . . . . . . 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 5883 . . . . . . . . . . 11  |-  ( Cs  S )  e.  _V
302, 29eqeltri 2353 . . . . . . . . . 10  |-  D  e. 
_V
3130a1i 10 . . . . . . . . 9  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  D  e.  _V )
32 ovex 5883 . . . . . . . . . 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 13774 . . . . . . . 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 13724 . . . . . . . . 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 13772 . . . . . . . . 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 3212 . . . . . . 7  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  ( D  Func  D
)  C_  ( D  Func  C ) )
3938, 7sseldd 3181 . . . . . 6  |-  ( ( C  e.  Cat  /\  S  e.  V )  ->  I  e.  ( D 
Func  C ) )
409, 39eqeltrrd 2358 . . . . 5  |-  ( ( C  e.  Cat  /\  S  e.  V )  -> 
<. ( 1st `  I
) ,  ( 2nd `  I ) >.  e.  ( D  Func  C )
)
41 df-br 4024 . . . . 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 5511 . . . . . 6  |-  (  _I  |`  ( x (  Hom  `  D ) y ) ) : ( x (  Hom  `  D
) y ) -1-1-onto-> ( x (  Hom  `  D
) y )
44 eqid 2283 . . . . . . . 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 2283 . . . . . . . 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 13751 . . . . . . 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 2284 . . . . . . 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 2283 . . . . . . . . . 10  |-  (  Hom  `  C )  =  (  Hom  `  C )
522, 51resshom 13323 . . . . . . . . 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 13754 . . . . . . . 8  |-  ( ( ( C  e.  Cat  /\  S  e.  V )  /\  ( x  e.  ( Base `  D
)  /\  y  e.  ( Base `  D )
) )  ->  (
( 1st `  I
) `  x )  =  x )
555, 44, 45, 48idfu1 13754 . . . . . . . 8  |-  ( ( ( C  e.  Cat  /\  S  e.  V )  /\  ( x  e.  ( Base `  D
)  /\  y  e.  ( Base `  D )
) )  ->  (
( 1st `  I
) `  y )  =  y )
5653, 54, 55oveq123d 5879 . . . . . . 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 5469 . . . . . 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 2635 . . . 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 13790 . . . 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 4024 . . 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 2357 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 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    i^i cin 3151    C_ wss 3152   <.cop 3643   class class class wbr 4023    _I cid 4304    X. cxp 4687    |` cres 4691   Rel wrel 4694   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   Basecbs 13148   ↾s cress 13149    Hom chom 13219   Catccat 13566    Homf chomf 13568  compfccomf 13569    |`cat cresc 13685  Subcatcsubc 13686    Func cfunc 13728  idfunccidfu 13729   Full cful 13776   Faith cfth 13777
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-hom 13232  df-cco 13233  df-cat 13570  df-cid 13571  df-homf 13572  df-comf 13573  df-ssc 13687  df-resc 13688  df-subc 13689  df-func 13732  df-idfu 13733  df-full 13778  df-fth 13779
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