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Theorem fthf1 15420
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 15418 . . . 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 462 . . 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 17 . 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 463 . . . 4  |-  ( (
ph  /\  x  =  X )  ->  Y  e.  B )
11 simplr 754 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  x  =  X )
12 simpr 459 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  y  =  Y )
1311, 12oveq12d 6250 . . . . 5  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  (
x G y )  =  ( X G Y ) )
1411, 12oveq12d 6250 . . . . 5  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  (
x H y )  =  ( X H Y ) )
1511fveq2d 5807 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  ( F `  x )  =  ( F `  X ) )
1612fveq2d 5807 . . . . . 6  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  ( F `  y )  =  ( F `  Y ) )
1715, 16oveq12d 6250 . . . . 5  |-  ( ( ( ph  /\  x  =  X )  /\  y  =  Y )  ->  (
( F `  x
) J ( F `
 y ) )  =  ( ( F `
 X ) J ( F `  Y
) ) )
1813, 14, 17f1eq123d 5748 . . . 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 3160 . . 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 3158 . 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 367    = wceq 1403    e. wcel 1840   A.wral 2751   class class class wbr 4392   -1-1->wf1 5520   ` cfv 5523  (class class class)co 6232   Basecbs 14731   Hom chom 14810    Func cfunc 15357   Faith cfth 15406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-1st 6736  df-2nd 6737  df-map 7377  df-ixp 7426  df-func 15361  df-fth 15408
This theorem is referenced by:  fthi  15421  ffthf1o  15422  fthoppc  15426  cofth  15438
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