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Theorem fthfunc 15133
Description: A faithful functor is a functor. (Contributed by Mario Carneiro, 26-Jan-2017.)
Assertion
Ref Expression
fthfunc  |-  ( C Faith 
D )  C_  ( C  Func  D )

Proof of Theorem fthfunc
Dummy variables  c 
d  f  g  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 6290 . . . 4  |-  ( c  =  C  ->  (
c Faith  d )  =  ( C Faith  d ) )
2 oveq1 6290 . . . 4  |-  ( c  =  C  ->  (
c  Func  d )  =  ( C  Func  d ) )
31, 2sseq12d 3533 . . 3  |-  ( c  =  C  ->  (
( c Faith  d ) 
C_  ( c  Func  d )  <->  ( C Faith  d
)  C_  ( C  Func  d ) ) )
4 oveq2 6291 . . . 4  |-  ( d  =  D  ->  ( C Faith  d )  =  ( C Faith  D ) )
5 oveq2 6291 . . . 4  |-  ( d  =  D  ->  ( C  Func  d )  =  ( C  Func  D
) )
64, 5sseq12d 3533 . . 3  |-  ( d  =  D  ->  (
( C Faith  d )  C_  ( C  Func  d
)  <->  ( C Faith  D
)  C_  ( C  Func  D ) ) )
7 ovex 6308 . . . . . 6  |-  ( c 
Func  d )  e. 
_V
8 simpl 457 . . . . . . . 8  |-  ( ( f ( c  Func  d ) g  /\  A. x  e.  ( Base `  c ) A. y  e.  ( Base `  c
) Fun  `' (
x g y ) )  ->  f (
c  Func  d )
g )
98ssopab2i 4775 . . . . . . 7  |-  { <. f ,  g >.  |  ( f ( c  Func  d ) g  /\  A. x  e.  ( Base `  c ) A. y  e.  ( Base `  c
) Fun  `' (
x g y ) ) }  C_  { <. f ,  g >.  |  f ( c  Func  d
) g }
10 opabss 4508 . . . . . . 7  |-  { <. f ,  g >.  |  f ( c  Func  d
) g }  C_  ( c  Func  d
)
119, 10sstri 3513 . . . . . 6  |-  { <. f ,  g >.  |  ( f ( c  Func  d ) g  /\  A. x  e.  ( Base `  c ) A. y  e.  ( Base `  c
) Fun  `' (
x g y ) ) }  C_  (
c  Func  d )
127, 11ssexi 4592 . . . . 5  |-  { <. f ,  g >.  |  ( f ( c  Func  d ) g  /\  A. x  e.  ( Base `  c ) A. y  e.  ( Base `  c
) Fun  `' (
x g y ) ) }  e.  _V
13 df-fth 15131 . . . . . 6  |- Faith  =  ( c  e.  Cat , 
d  e.  Cat  |->  {
<. f ,  g >.  |  ( f ( c  Func  d )
g  /\  A. x  e.  ( Base `  c
) A. y  e.  ( Base `  c
) Fun  `' (
x g y ) ) } )
1413ovmpt4g 6408 . . . . 5  |-  ( ( c  e.  Cat  /\  d  e.  Cat  /\  { <. f ,  g >.  |  ( f ( c  Func  d )
g  /\  A. x  e.  ( Base `  c
) A. y  e.  ( Base `  c
) Fun  `' (
x g y ) ) }  e.  _V )  ->  ( c Faith  d
)  =  { <. f ,  g >.  |  ( f ( c  Func  d ) g  /\  A. x  e.  ( Base `  c ) A. y  e.  ( Base `  c
) Fun  `' (
x g y ) ) } )
1512, 14mp3an3 1313 . . . 4  |-  ( ( c  e.  Cat  /\  d  e.  Cat )  ->  ( c Faith  d )  =  { <. f ,  g >.  |  ( f ( c  Func  d ) g  /\  A. x  e.  ( Base `  c ) A. y  e.  ( Base `  c
) Fun  `' (
x g y ) ) } )
1615, 11syl6eqss 3554 . . 3  |-  ( ( c  e.  Cat  /\  d  e.  Cat )  ->  ( c Faith  d ) 
C_  ( c  Func  d ) )
173, 6, 16vtocl2ga 3179 . 2  |-  ( ( C  e.  Cat  /\  D  e.  Cat )  ->  ( C Faith  D ) 
C_  ( C  Func  D ) )
1813mpt2ndm0 6499 . . 3  |-  ( -.  ( C  e.  Cat  /\  D  e.  Cat )  ->  ( C Faith  D )  =  (/) )
19 0ss 3814 . . 3  |-  (/)  C_  ( C  Func  D )
2018, 19syl6eqss 3554 . 2  |-  ( -.  ( C  e.  Cat  /\  D  e.  Cat )  ->  ( C Faith  D ) 
C_  ( C  Func  D ) )
2117, 20pm2.61i 164 1  |-  ( C Faith 
D )  C_  ( C  Func  D )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113    C_ wss 3476   (/)c0 3785   class class class wbr 4447   {copab 4504   `'ccnv 4998   Fun wfun 5581   ` cfv 5587  (class class class)co 6283   Basecbs 14489   Catccat 14918    Func cfunc 15080   Faith cfth 15129
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-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  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-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5550  df-fun 5589  df-fv 5595  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-fth 15131
This theorem is referenced by:  relfth  15135  isfth  15140  fthoppc  15149  fthsect  15151  fthinv  15152  fthmon  15153  fthepi  15154  ffthiso  15155  cofth  15161
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