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Theorem fthmon 15165
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 2441 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
3 eqid 2441 . . . . . 6  |-  ( Hom  `  D )  =  ( Hom  `  D )
4 eqid 2441 . . . . . 6  |-  (comp `  D )  =  (comp `  D )
5 fthmon.n . . . . . 6  |-  N  =  (Mono `  D )
6 fthmon.f . . . . . . . . . . 11  |-  ( ph  ->  F ( C Faith  D
) G )
7 fthfunc 15145 . . . . . . . . . . . 12  |-  ( C Faith 
D )  C_  ( C  Func  D )
87ssbri 4475 . . . . . . . . . . 11  |-  ( F ( C Faith  D ) G  ->  F ( C  Func  D ) G )
96, 8syl 16 . . . . . . . . . 10  |-  ( ph  ->  F ( C  Func  D ) G )
10 df-br 4434 . . . . . . . . . 10  |-  ( F ( C  Func  D
) G  <->  <. F ,  G >.  e.  ( C 
Func  D ) )
119, 10sylib 196 . . . . . . . . 9  |-  ( ph  -> 
<. F ,  G >.  e.  ( C  Func  D
) )
12 funcrcl 15101 . . . . . . . . 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 463 . . . . . . 7  |-  ( ph  ->  D  e.  Cat )
1514adantr 465 . . . . . 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 465 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  ->  F ( C  Func  D ) G )
1816, 2, 17funcf1 15104 . . . . . . 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 465 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  ->  X  e.  B )
2118, 20ffvelrnd 6013 . . . . . 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 465 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  ->  Y  e.  B )
2418, 23ffvelrnd 6013 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  -> 
( F `  Y
)  e.  ( Base `  D ) )
25 simpr1 1001 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  -> 
z  e.  B )
2618, 25ffvelrnd 6013 . . . . . 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 465 . . . . . 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 15106 . . . . . . 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 1002 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  -> 
f  e.  ( z H X ) )
3230, 31ffvelrnd 6013 . . . . . 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 1003 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  -> 
g  e.  ( z H X ) )
3430, 33ffvelrnd 6013 . . . . . 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 15003 . . . . 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 2441 . . . . . . . 8  |-  (comp `  C )  =  (comp `  C )
371adantr 465 . . . . . . . 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 15109 . . . . . . 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 15109 . . . . . . 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 2463 . . . . . 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 465 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  ->  F ( C Faith  D
) G )
4213simpld 459 . . . . . . . . 9  |-  ( ph  ->  C  e.  Cat )
4342adantr 465 . . . . . . . 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 14954 . . . . . . 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 14954 . . . . . . 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 15156 . . . . . 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 255 . . . . 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 15156 . . . . 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 283 . . . 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 207 . . 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 2863 . 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 15001 . 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 920 1  |-  ( ph  ->  R  e.  ( X M Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802   A.wral 2791   <.cop 4016   class class class wbr 4433   ` cfv 5574  (class class class)co 6277   Basecbs 14504   Hom chom 14580  compcco 14581   Catccat 14933  Monocmon 14995    Func cfunc 15092   Faith cfth 15141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6781  df-2nd 6782  df-map 7420  df-ixp 7468  df-cat 14937  df-mon 14997  df-func 15096  df-fth 15143
This theorem is referenced by:  fthepi  15166
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