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Theorem fthi 14115
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 )
fthi.r  |-  ( ph  ->  R  e.  ( X H Y ) )
fthi.s  |-  ( ph  ->  S  e.  ( X H Y ) )
Assertion
Ref Expression
fthi  |-  ( ph  ->  ( ( ( X G Y ) `  R )  =  ( ( X G Y ) `  S )  <-> 
R  =  S ) )

Proof of Theorem fthi
StepHypRef Expression
1 isfth.b . . 3  |-  B  =  ( Base `  C
)
2 isfth.h . . 3  |-  H  =  (  Hom  `  C
)
3 isfth.j . . 3  |-  J  =  (  Hom  `  D
)
4 fthf1.f . . 3  |-  ( ph  ->  F ( C Faith  D
) G )
5 fthf1.x . . 3  |-  ( ph  ->  X  e.  B )
6 fthf1.y . . 3  |-  ( ph  ->  Y  e.  B )
71, 2, 3, 4, 5, 6fthf1 14114 . 2  |-  ( ph  ->  ( X G Y ) : ( X H Y ) -1-1-> ( ( F `  X
) J ( F `
 Y ) ) )
8 fthi.r . 2  |-  ( ph  ->  R  e.  ( X H Y ) )
9 fthi.s . 2  |-  ( ph  ->  S  e.  ( X H Y ) )
10 f1fveq 6008 . 2  |-  ( ( ( X G Y ) : ( X H Y ) -1-1-> ( ( F `  X
) J ( F `
 Y ) )  /\  ( R  e.  ( X H Y )  /\  S  e.  ( X H Y ) ) )  -> 
( ( ( X G Y ) `  R )  =  ( ( X G Y ) `  S )  <-> 
R  =  S ) )
117, 8, 9, 10syl12anc 1182 1  |-  ( ph  ->  ( ( ( X G Y ) `  R )  =  ( ( X G Y ) `  S )  <-> 
R  =  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725   class class class wbr 4212   -1-1->wf1 5451   ` cfv 5454  (class class class)co 6081   Basecbs 13469    Hom chom 13540   Faith cfth 14100
This theorem is referenced by:  fthsect  14122  fthmon  14124
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-map 7020  df-ixp 7064  df-func 14055  df-fth 14102
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