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Theorem fthfunc 15763
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 6312 . . . 4  |-  ( c  =  C  ->  (
c Faith  d )  =  ( C Faith  d ) )
2 oveq1 6312 . . . 4  |-  ( c  =  C  ->  (
c  Func  d )  =  ( C  Func  d ) )
31, 2sseq12d 3499 . . 3  |-  ( c  =  C  ->  (
( c Faith  d ) 
C_  ( c  Func  d )  <->  ( C Faith  d
)  C_  ( C  Func  d ) ) )
4 oveq2 6313 . . . 4  |-  ( d  =  D  ->  ( C Faith  d )  =  ( C Faith  D ) )
5 oveq2 6313 . . . 4  |-  ( d  =  D  ->  ( C  Func  d )  =  ( C  Func  D
) )
64, 5sseq12d 3499 . . 3  |-  ( d  =  D  ->  (
( C Faith  d )  C_  ( C  Func  d
)  <->  ( C Faith  D
)  C_  ( C  Func  D ) ) )
7 ovex 6333 . . . . . 6  |-  ( c 
Func  d )  e. 
_V
8 simpl 458 . . . . . . . 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 4749 . . . . . . 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 4487 . . . . . . 7  |-  { <. f ,  g >.  |  f ( c  Func  d
) g }  C_  ( c  Func  d
)
119, 10sstri 3479 . . . . . 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 4570 . . . . 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 15761 . . . . . 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 6433 . . . . 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 1349 . . . 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 3520 . . 3  |-  ( ( c  e.  Cat  /\  d  e.  Cat )  ->  ( c Faith  d ) 
C_  ( c  Func  d ) )
173, 6, 16vtocl2ga 3153 . 2  |-  ( ( C  e.  Cat  /\  D  e.  Cat )  ->  ( C Faith  D ) 
C_  ( C  Func  D ) )
1813mpt2ndm0 6524 . . 3  |-  ( -.  ( C  e.  Cat  /\  D  e.  Cat )  ->  ( C Faith  D )  =  (/) )
19 0ss 3797 . . 3  |-  (/)  C_  ( C  Func  D )
2018, 19syl6eqss 3520 . 2  |-  ( -.  ( C  e.  Cat  /\  D  e.  Cat )  ->  ( C Faith  D ) 
C_  ( C  Func  D ) )
2117, 20pm2.61i 167 1  |-  ( C Faith 
D )  C_  ( C  Func  D )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782   _Vcvv 3087    C_ wss 3442   (/)c0 3767   class class class wbr 4426   {copab 4483   `'ccnv 4853   Fun wfun 5595   ` cfv 5601  (class class class)co 6305   Basecbs 15084   Catccat 15521    Func cfunc 15710   Faith cfth 15759
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 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-fth 15761
This theorem is referenced by:  relfth  15765  isfth  15770  fthoppc  15779  fthsect  15781  fthinv  15782  fthmon  15783  fthepi  15784  ffthiso  15785  cofth  15791  inclfusubc  38624
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