MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fthmon Unicode version

Theorem fthmon 13801
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 2283 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
3 eqid 2283 . . . . . 6  |-  (  Hom  `  D )  =  (  Hom  `  D )
4 eqid 2283 . . . . . 6  |-  (comp `  D )  =  (comp `  D )
5 fthmon.n . . . . . 6  |-  N  =  (Mono `  D )
6 fthmon.f . . . . . . . . . . 11  |-  ( ph  ->  F ( C Faith  D
) G )
7 fthfunc 13781 . . . . . . . . . . . 12  |-  ( C Faith 
D )  C_  ( C  Func  D )
87ssbri 4065 . . . . . . . . . . 11  |-  ( F ( C Faith  D ) G  ->  F ( C  Func  D ) G )
96, 8syl 15 . . . . . . . . . 10  |-  ( ph  ->  F ( C  Func  D ) G )
10 df-br 4024 . . . . . . . . . 10  |-  ( F ( C  Func  D
) G  <->  <. F ,  G >.  e.  ( C 
Func  D ) )
119, 10sylib 188 . . . . . . . . 9  |-  ( ph  -> 
<. F ,  G >.  e.  ( C  Func  D
) )
12 funcrcl 13737 . . . . . . . . 9  |-  ( <. F ,  G >.  e.  ( C  Func  D
)  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
1311, 12syl 15 . . . . . . . 8  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
1413simprd 449 . . . . . . 7  |-  ( ph  ->  D  e.  Cat )
1514adantr 451 . . . . . 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 451 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  ->  F ( C  Func  D ) G )
1816, 2, 17funcf1 13740 . . . . . . 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 451 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  ->  X  e.  B )
2118, 20ffvelrnd 5666 . . . . . 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 451 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  ->  Y  e.  B )
2418, 23ffvelrnd 5666 . . . . . 6  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  -> 
( F `  Y
)  e.  ( Base `  D ) )
25 simpr1 961 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  -> 
z  e.  B )
2618, 25ffvelrnd 5666 . . . . . 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 451 . . . . . 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 13742 . . . . . . 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 962 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  -> 
f  e.  ( z H X ) )
3230, 31ffvelrnd 5666 . . . . . 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 963 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  -> 
g  e.  ( z H X ) )
3430, 33ffvelrnd 5666 . . . . . 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 13639 . . . . 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 2283 . . . . . . . 8  |-  (comp `  C )  =  (comp `  C )
371adantr 451 . . . . . . . 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 13745 . . . . . . 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 13745 . . . . . . 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 2297 . . . . . 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 451 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  B  /\  f  e.  ( z H X )  /\  g  e.  ( z H X ) ) )  ->  F ( C Faith  D
) G )
4213simpld 445 . . . . . . . . 9  |-  ( ph  ->  C  e.  Cat )
4342adantr 451 . . . . . . . 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 13587 . . . . . . 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 13587 . . . . . . 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 13792 . . . . . 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 246 . . . . 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 13792 . . . . 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 274 . . . 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 198 . . 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 2636 . 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 13637 . 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 888 1  |-  ( ph  ->  R  e.  ( X M Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   <.cop 3643   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148    Hom chom 13219  compcco 13220   Catccat 13566  Monocmon 13631    Func cfunc 13728   Faith cfth 13777
This theorem is referenced by:  fthepi  13802
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-cat 13570  df-mon 13633  df-func 13732  df-fth 13779
  Copyright terms: Public domain W3C validator