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Theorem fthi 13792
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 13791 . 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 5786 . 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 1180 1  |-  ( ph  ->  ( ( ( X G Y ) `  R )  =  ( ( X G Y ) `  S )  <-> 
R  =  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   class class class wbr 4023   -1-1->wf1 5252   ` cfv 5255  (class class class)co 5858   Basecbs 13148    Hom chom 13219   Faith cfth 13777
This theorem is referenced by:  fthsect  13799  fthmon  13801
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-ixp 6818  df-func 13732  df-fth 13779
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