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Theorem fthf1 14116
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 14114 . . . 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 452 . . 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 453 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  Y  e.  B )
11 simplr 733 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  x  =  X )
12 simpr 449 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  y  =  Y )
1311, 12oveq12d 6101 . . . . 5  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  (
x G y )  =  ( X G Y ) )
1411, 12oveq12d 6101 . . . . 5  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  (
x H y )  =  ( X H Y ) )
1511fveq2d 5734 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  ( F `  x )  =  ( F `  X ) )
1612fveq2d 5734 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  ( F `  y )  =  ( F `  Y ) )
1715, 16oveq12d 6101 . . . . 5  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  (
( F `  x
) J ( F `
 y ) )  =  ( ( F `
 X ) J ( F `  Y
) ) )
1813, 14, 17f1eq123d 5671 . . . 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 3057 . . 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 3055 . 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 set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   class class class wbr 4214   -1-1->wf1 5453   ` cfv 5456  (class class class)co 6083   Basecbs 13471    Hom chom 13542    Func cfunc 14053   Faith cfth 14102
This theorem is referenced by:  fthi  14117  ffthf1o  14118  fthoppc  14122  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-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
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-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-map 7022  df-ixp 7066  df-func 14057  df-fth 14104
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