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Theorem fthfunc 14106
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 6090 . . . 4  |-  ( c  =  C  ->  (
c Faith  d )  =  ( C Faith  d ) )
2 oveq1 6090 . . . 4  |-  ( c  =  C  ->  (
c  Func  d )  =  ( C  Func  d ) )
31, 2sseq12d 3379 . . 3  |-  ( c  =  C  ->  (
( c Faith  d ) 
C_  ( c  Func  d )  <->  ( C Faith  d
)  C_  ( C  Func  d ) ) )
4 oveq2 6091 . . . 4  |-  ( d  =  D  ->  ( C Faith  d )  =  ( C Faith  D ) )
5 oveq2 6091 . . . 4  |-  ( d  =  D  ->  ( C  Func  d )  =  ( C  Func  D
) )
64, 5sseq12d 3379 . . 3  |-  ( d  =  D  ->  (
( C Faith  d )  C_  ( C  Func  d
)  <->  ( C Faith  D
)  C_  ( C  Func  D ) ) )
7 ovex 6108 . . . . . 6  |-  ( c 
Func  d )  e. 
_V
8 simpl 445 . . . . . . . 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 4484 . . . . . . 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 4271 . . . . . . 7  |-  { <. f ,  g >.  |  f ( c  Func  d
) g }  C_  ( c  Func  d
)
119, 10sstri 3359 . . . . . 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 4350 . . . . 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 14104 . . . . . 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 6198 . . . . 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 1269 . . . 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 3400 . . 3  |-  ( ( c  e.  Cat  /\  d  e.  Cat )  ->  ( c Faith  d ) 
C_  ( c  Func  d ) )
173, 6, 16vtocl2ga 3021 . 2  |-  ( ( C  e.  Cat  /\  D  e.  Cat )  ->  ( C Faith  D ) 
C_  ( C  Func  D ) )
1813mpt2ndm0 6475 . . 3  |-  ( -.  ( C  e.  Cat  /\  D  e.  Cat )  ->  ( C Faith  D )  =  (/) )
19 0ss 3658 . . 3  |-  (/)  C_  ( C  Func  D )
2018, 19syl6eqss 3400 . 2  |-  ( -.  ( C  e.  Cat  /\  D  e.  Cat )  ->  ( C Faith  D ) 
C_  ( C  Func  D ) )
2117, 20pm2.61i 159 1  |-  ( C Faith 
D )  C_  ( C  Func  D )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958    C_ wss 3322   (/)c0 3630   class class class wbr 4214   {copab 4267   `'ccnv 4879   Fun wfun 5450   ` cfv 5456  (class class class)co 6083   Basecbs 13471   Catccat 13891    Func cfunc 14053   Faith cfth 14102
This theorem is referenced by:  relfth  14108  isfth  14113  fthoppc  14122  fthsect  14124  fthinv  14125  fthmon  14126  fthepi  14127  ffthiso  14128  cofth  14134
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-fth 14104
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