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Theorem fthmon 14129
Description: A faithful functor reflects monomorphisms. (Contributed by Mario Carneiro, 27-Jan-2017.)
Hypotheses
Ref Expression
fthmon.b  |-  B  =  ( Base `  C
)
fthmon.h  |-  H  =  (  Hom  `  C
)
fthmon.f  |-  ( ph  ->  F ( C Faith  D
) G )
fthmon.x  |-  ( ph  ->  X  e.  B )
fthmon.y  |-  ( ph  ->  Y  e.  B )
fthmon.r  |-  ( ph  ->  R  e.  ( X H Y ) )
fthmon.m  |-  M  =  (Mono `  C )
fthmon.n  |-  N  =  (Mono `  D )
fthmon.1  |-  ( ph  ->  ( ( X G Y ) `  R
)  e.  ( ( F `  X ) N ( F `  Y ) ) )
Assertion
Ref Expression
fthmon  |-  ( ph  ->  R  e.  ( X M Y ) )

Proof of Theorem fthmon
Dummy variables  f 
g  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fthmon.r . 2  |-  ( ph  ->  R  e.  ( X H Y ) )
2 eqid 2438 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
3 eqid 2438 . . . . . 6  |-  (  Hom  `  D )  =  (  Hom  `  D )
4 eqid 2438 . . . . . 6  |-  (comp `  D )  =  (comp `  D )
5 fthmon.n . . . . . 6  |-  N  =  (Mono `  D )
6 fthmon.f . . . . . . . . . . 11  |-  ( ph  ->  F ( C Faith  D
) G )
7 fthfunc 14109 . . . . . . . . . . . 12  |-  ( C Faith 
D )  C_  ( C  Func  D )
87ssbri 4257 . . . . . . . . . . 11  |-  ( F ( C Faith  D ) G  ->  F ( C  Func  D ) G )
96, 8syl 16 . . . . . . . . . 10  |-  ( ph  ->  F ( C  Func  D ) G )
10 df-br 4216 . . . . . . . . . 10  |-  ( F ( C  Func  D
) G  <->  <. F ,  G >.  e.  ( C 
Func  D ) )
119, 10sylib 190 . . . . . . . . 9  |-  ( ph  -> 
<. F ,  G >.  e.  ( C  Func  D
) )
12 funcrcl 14065 . . . . . . . . 9  |-  ( <. F ,  G >.  e.  ( C  Func  D
)  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
1311, 12syl 16 . . . . . . . 8  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
1413simprd 451 . . . . . . 7  |-  ( ph  ->  D  e.  Cat )
1514adantr 453 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  ->  D  e.  Cat )
16 fthmon.b . . . . . . . 8  |-  B  =  ( Base `  C
)
179adantr 453 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  ->  F ( C  Func  D ) G )
1816, 2, 17funcf1 14068 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  ->  F : B --> ( Base `  D ) )
19 fthmon.x . . . . . . . 8  |-  ( ph  ->  X  e.  B )
2019adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  ->  X  e.  B )
2118, 20ffvelrnd 5874 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  -> 
( F `  X
)  e.  ( Base `  D ) )
22 fthmon.y . . . . . . . 8  |-  ( ph  ->  Y  e.  B )
2322adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  ->  Y  e.  B )
2418, 23ffvelrnd 5874 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  -> 
( F `  Y
)  e.  ( Base `  D ) )
25 simpr1 964 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  -> 
z  e.  B )
2618, 25ffvelrnd 5874 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  -> 
( F `  z
)  e.  ( Base `  D ) )
27 fthmon.1 . . . . . . 7  |-  ( ph  ->  ( ( X G Y ) `  R
)  e.  ( ( F `  X ) N ( F `  Y ) ) )
2827adantr 453 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  -> 
( ( X G Y ) `  R
)  e.  ( ( F `  X ) N ( F `  Y ) ) )
29 fthmon.h . . . . . . . 8  |-  H  =  (  Hom  `  C
)
3016, 29, 3, 17, 25, 20funcf2 14070 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  -> 
( z G X ) : ( z H X ) --> ( ( F `  z
) (  Hom  `  D
) ( F `  X ) ) )
31 simpr2 965 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  -> 
f  e.  ( z H X ) )
3230, 31ffvelrnd 5874 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  -> 
( ( z G X ) `  f
)  e.  ( ( F `  z ) (  Hom  `  D
) ( F `  X ) ) )
33 simpr3 966 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  -> 
g  e.  ( z H X ) )
3430, 33ffvelrnd 5874 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  -> 
( ( z G X ) `  g
)  e.  ( ( F `  z ) (  Hom  `  D
) ( F `  X ) ) )
352, 3, 4, 5, 15, 21, 24, 26, 28, 32, 34moni 13967 . . . . 5  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  -> 
( ( ( ( X G Y ) `
 R ) (
<. ( F `  z
) ,  ( F `
 X ) >.
(comp `  D )
( F `  Y
) ) ( ( z G X ) `
 f ) )  =  ( ( ( X G Y ) `
 R ) (
<. ( F `  z
) ,  ( F `
 X ) >.
(comp `  D )
( F `  Y
) ) ( ( z G X ) `
 g ) )  <-> 
( ( z G X ) `  f
)  =  ( ( z G X ) `
 g ) ) )
36 eqid 2438 . . . . . . . 8  |-  (comp `  C )  =  (comp `  C )
371adantr 453 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  ->  R  e.  ( X H Y ) )
3816, 29, 36, 4, 17, 25, 20, 23, 31, 37funcco 14073 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  -> 
( ( z G Y ) `  ( R ( <. z ,  X >. (comp `  C
) Y ) f ) )  =  ( ( ( X G Y ) `  R
) ( <. ( F `  z ) ,  ( F `  X ) >. (comp `  D ) ( F `
 Y ) ) ( ( z G X ) `  f
) ) )
3916, 29, 36, 4, 17, 25, 20, 23, 33, 37funcco 14073 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  -> 
( ( z G Y ) `  ( R ( <. z ,  X >. (comp `  C
) Y ) g ) )  =  ( ( ( X G Y ) `  R
) ( <. ( F `  z ) ,  ( F `  X ) >. (comp `  D ) ( F `
 Y ) ) ( ( z G X ) `  g
) ) )
4038, 39eqeq12d 2452 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  -> 
( ( ( z G Y ) `  ( R ( <. z ,  X >. (comp `  C
) Y ) f ) )  =  ( ( z G Y ) `  ( R ( <. z ,  X >. (comp `  C ) Y ) g ) )  <->  ( ( ( X G Y ) `
 R ) (
<. ( F `  z
) ,  ( F `
 X ) >.
(comp `  D )
( F `  Y
) ) ( ( z G X ) `
 f ) )  =  ( ( ( X G Y ) `
 R ) (
<. ( F `  z
) ,  ( F `
 X ) >.
(comp `  D )
( F `  Y
) ) ( ( z G X ) `
 g ) ) ) )
416adantr 453 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  ->  F ( C Faith  D
) G )
4213simpld 447 . . . . . . . . 9  |-  ( ph  ->  C  e.  Cat )
4342adantr 453 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  ->  C  e.  Cat )
4416, 29, 36, 43, 25, 20, 23, 31, 37catcocl 13915 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  -> 
( R ( <.
z ,  X >. (comp `  C ) Y ) f )  e.  ( z H Y ) )
4516, 29, 36, 43, 25, 20, 23, 33, 37catcocl 13915 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  -> 
( R ( <.
z ,  X >. (comp `  C ) Y ) g )  e.  ( z H Y ) )
4616, 29, 3, 41, 25, 23, 44, 45fthi 14120 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  -> 
( ( ( z G Y ) `  ( R ( <. z ,  X >. (comp `  C
) Y ) f ) )  =  ( ( z G Y ) `  ( R ( <. z ,  X >. (comp `  C ) Y ) g ) )  <->  ( R (
<. z ,  X >. (comp `  C ) Y ) f )  =  ( R ( <. z ,  X >. (comp `  C
) Y ) g ) ) )
4740, 46bitr3d 248 . . . . 5  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  -> 
( ( ( ( X G Y ) `
 R ) (
<. ( F `  z
) ,  ( F `
 X ) >.
(comp `  D )
( F `  Y
) ) ( ( z G X ) `
 f ) )  =  ( ( ( X G Y ) `
 R ) (
<. ( F `  z
) ,  ( F `
 X ) >.
(comp `  D )
( F `  Y
) ) ( ( z G X ) `
 g ) )  <-> 
( R ( <.
z ,  X >. (comp `  C ) Y ) f )  =  ( R ( <. z ,  X >. (comp `  C
) Y ) g ) ) )
4816, 29, 3, 41, 25, 20, 31, 33fthi 14120 . . . . 5  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  -> 
( ( ( z G X ) `  f )  =  ( ( z G X ) `  g )  <-> 
f  =  g ) )
4935, 47, 483bitr3d 276 . . . 4  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  -> 
( ( R (
<. z ,  X >. (comp `  C ) Y ) f )  =  ( R ( <. z ,  X >. (comp `  C
) Y ) g )  <->  f  =  g ) )
5049biimpd 200 . . 3  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  -> 
( ( R (
<. z ,  X >. (comp `  C ) Y ) f )  =  ( R ( <. z ,  X >. (comp `  C
) Y ) g )  ->  f  =  g ) )
5150ralrimivvva 2801 . 2  |-  ( ph  ->  A. z  e.  B  A. f  e.  (
z H X ) A. g  e.  ( z H X ) ( ( R (
<. z ,  X >. (comp `  C ) Y ) f )  =  ( R ( <. z ,  X >. (comp `  C
) Y ) g )  ->  f  =  g ) )
52 fthmon.m . . 3  |-  M  =  (Mono `  C )
5316, 29, 36, 52, 42, 19, 22ismon2 13965 . 2  |-  ( ph  ->  ( R  e.  ( X M Y )  <-> 
( R  e.  ( X H Y )  /\  A. z  e.  B  A. f  e.  ( z H X ) A. g  e.  ( z H X ) ( ( R ( <. z ,  X >. (comp `  C ) Y ) f )  =  ( R (
<. z ,  X >. (comp `  C ) Y ) g )  ->  f  =  g ) ) ) )
541, 51, 53mpbir2and 890 1  |-  ( ph  ->  R  e.  ( X M Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   <.cop 3819   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   Basecbs 13474    Hom chom 13545  compcco 13546   Catccat 13894  Monocmon 13959    Func cfunc 14056   Faith cfth 14105
This theorem is referenced by:  fthepi  14130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
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 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-map 7023  df-ixp 7067  df-cat 13898  df-mon 13961  df-func 14060  df-fth 14107
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