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

Theorem fthmon 13817
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 2296 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
3 eqid 2296 . . . . . 6  |-  (  Hom  `  D )  =  (  Hom  `  D )
4 eqid 2296 . . . . . 6  |-  (comp `  D )  =  (comp `  D )
5 fthmon.n . . . . . 6  |-  N  =  (Mono `  D )
6 fthmon.f . . . . . . . . . . 11  |-  ( ph  ->  F ( C Faith  D
) G )
7 fthfunc 13797 . . . . . . . . . . . 12  |-  ( C Faith 
D )  C_  ( C  Func  D )
87ssbri 4081 . . . . . . . . . . 11  |-  ( F ( C Faith  D ) G  ->  F ( C  Func  D ) G )
96, 8syl 15 . . . . . . . . . 10  |-  ( ph  ->  F ( C  Func  D ) G )
10 df-br 4040 . . . . . . . . . 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 13753 . . . . . . . . 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 13756 . . . . . . 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 5682 . . . . . 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 5682 . . . . . 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 5682 . . . . . 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 13758 . . . . . . 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 5682 . . . . . 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 5682 . . . . . 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 13655 . . . . 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 2296 . . . . . . . 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 13761 . . . . . . 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 13761 . . . . . . 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 2310 . . . . . 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 13603 . . . . . . 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 13603 . . . . . . 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 13808 . . . . . 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 13808 . . . . 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 2649 . 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 13653 . 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 1632    e. wcel 1696   A.wral 2556   <.cop 3656   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164    Hom chom 13235  compcco 13236   Catccat 13582  Monocmon 13647    Func cfunc 13744   Faith cfth 13793
This theorem is referenced by:  fthepi  13818
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-map 6790  df-ixp 6834  df-cat 13586  df-mon 13649  df-func 13748  df-fth 13795
  Copyright terms: Public domain W3C validator