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Theorem fthf1 14832
Description: The morphism map of a faithful functor is an injection. (Contributed by Mario Carneiro, 27-Jan-2017.)
Hypotheses
Ref Expression
isfth.b  |-  B  =  ( Base `  C
)
isfth.h  |-  H  =  ( Hom  `  C
)
isfth.j  |-  J  =  ( Hom  `  D
)
fthf1.f  |-  ( ph  ->  F ( C Faith  D
) G )
fthf1.x  |-  ( ph  ->  X  e.  B )
fthf1.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
fthf1  |-  ( ph  ->  ( X G Y ) : ( X H Y ) -1-1-> ( ( F `  X
) J ( F `
 Y ) ) )

Proof of Theorem fthf1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fthf1.f . . 3  |-  ( ph  ->  F ( C Faith  D
) G )
2 isfth.b . . . . 5  |-  B  =  ( Base `  C
)
3 isfth.h . . . . 5  |-  H  =  ( Hom  `  C
)
4 isfth.j . . . . 5  |-  J  =  ( Hom  `  D
)
52, 3, 4isfth2 14830 . . . 4  |-  ( F ( C Faith  D ) G  <->  ( F ( C  Func  D ) G  /\  A. x  e.  B  A. y  e.  B  ( x G y ) : ( x H y )
-1-1-> ( ( F `  x ) J ( F `  y ) ) ) )
65simprbi 464 . . 3  |-  ( F ( C Faith  D ) G  ->  A. x  e.  B  A. y  e.  B  ( x G y ) : ( x H y ) -1-1-> ( ( F `
 x ) J ( F `  y
) ) )
71, 6syl 16 . 2  |-  ( ph  ->  A. x  e.  B  A. y  e.  B  ( x G y ) : ( x H y ) -1-1-> ( ( F `  x
) J ( F `
 y ) ) )
8 fthf1.x . . 3  |-  ( ph  ->  X  e.  B )
9 fthf1.y . . . . 5  |-  ( ph  ->  Y  e.  B )
109adantr 465 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  Y  e.  B )
11 simplr 754 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  x  =  X )
12 simpr 461 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  y  =  Y )
1311, 12oveq12d 6114 . . . . 5  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  (
x G y )  =  ( X G Y ) )
1411, 12oveq12d 6114 . . . . 5  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  (
x H y )  =  ( X H Y ) )
1511fveq2d 5700 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  ( F `  x )  =  ( F `  X ) )
1612fveq2d 5700 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  ( F `  y )  =  ( F `  Y ) )
1715, 16oveq12d 6114 . . . . 5  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  (
( F `  x
) J ( F `
 y ) )  =  ( ( F `
 X ) J ( F `  Y
) ) )
1813, 14, 17f1eq123d 5641 . . . 4  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  (
( x G y ) : ( x H y ) -1-1-> ( ( F `  x
) J ( F `
 y ) )  <-> 
( X G Y ) : ( X H Y ) -1-1-> ( ( F `  X
) J ( F `
 Y ) ) ) )
1910, 18rspcdv 3081 . . 3  |-  ( (
ph  /\  x  =  X )  ->  ( A. y  e.  B  ( x G y ) : ( x H y ) -1-1-> ( ( F `  x
) J ( F `
 y ) )  ->  ( X G Y ) : ( X H Y )
-1-1-> ( ( F `  X ) J ( F `  Y ) ) ) )
208, 19rspcimdv 3079 . 2  |-  ( ph  ->  ( A. x  e.  B  A. y  e.  B  ( x G y ) : ( x H y )
-1-1-> ( ( F `  x ) J ( F `  y ) )  ->  ( X G Y ) : ( X H Y )
-1-1-> ( ( F `  X ) J ( F `  Y ) ) ) )
217, 20mpd 15 1  |-  ( ph  ->  ( X G Y ) : ( X H Y ) -1-1-> ( ( F `  X
) J ( F `
 Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2720   class class class wbr 4297   -1-1->wf1 5420   ` cfv 5423  (class class class)co 6096   Basecbs 14179   Hom chom 14254    Func cfunc 14769   Faith cfth 14818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-map 7221  df-ixp 7269  df-func 14773  df-fth 14820
This theorem is referenced by:  fthi  14833  ffthf1o  14834  fthoppc  14838  cofth  14850
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