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

Theorem monfval 13635
Description: Definition of a monomorphism in a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
ismon.b  |-  B  =  ( Base `  C
)
ismon.h  |-  H  =  (  Hom  `  C
)
ismon.o  |-  .x.  =  (comp `  C )
ismon.s  |-  M  =  (Mono `  C )
ismon.c  |-  ( ph  ->  C  e.  Cat )
Assertion
Ref Expression
monfval  |-  ( ph  ->  M  =  ( x  e.  B ,  y  e.  B  |->  { f  e.  ( x H y )  |  A. z  e.  B  Fun  `' ( g  e.  ( z H x ) 
|->  ( f ( <.
z ,  x >.  .x.  y ) g ) ) } ) )
Distinct variable groups:    f, g, x, y, z, B    ph, f,
g, x, y, z    C, f, g, x, y, z    f, H, g, x, y, z    .x. , f,
g, x, y, z   
f, M
Allowed substitution hints:    M( x, y, z, g)

Proof of Theorem monfval
Dummy variables  b 
c  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ismon.s . 2  |-  M  =  (Mono `  C )
2 ismon.c . . 3  |-  ( ph  ->  C  e.  Cat )
3 fvex 5539 . . . . . 6  |-  ( Base `  c )  e.  _V
43a1i 10 . . . . 5  |-  ( c  =  C  ->  ( Base `  c )  e. 
_V )
5 fveq2 5525 . . . . . 6  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
6 ismon.b . . . . . 6  |-  B  =  ( Base `  C
)
75, 6syl6eqr 2333 . . . . 5  |-  ( c  =  C  ->  ( Base `  c )  =  B )
8 fvex 5539 . . . . . . 7  |-  (  Hom  `  c )  e.  _V
98a1i 10 . . . . . 6  |-  ( ( c  =  C  /\  b  =  B )  ->  (  Hom  `  c
)  e.  _V )
10 simpl 443 . . . . . . . 8  |-  ( ( c  =  C  /\  b  =  B )  ->  c  =  C )
1110fveq2d 5529 . . . . . . 7  |-  ( ( c  =  C  /\  b  =  B )  ->  (  Hom  `  c
)  =  (  Hom  `  C ) )
12 ismon.h . . . . . . 7  |-  H  =  (  Hom  `  C
)
1311, 12syl6eqr 2333 . . . . . 6  |-  ( ( c  =  C  /\  b  =  B )  ->  (  Hom  `  c
)  =  H )
14 simplr 731 . . . . . . 7  |-  ( ( ( c  =  C  /\  b  =  B )  /\  h  =  H )  ->  b  =  B )
15 simpr 447 . . . . . . . . 9  |-  ( ( ( c  =  C  /\  b  =  B )  /\  h  =  H )  ->  h  =  H )
1615oveqd 5875 . . . . . . . 8  |-  ( ( ( c  =  C  /\  b  =  B )  /\  h  =  H )  ->  (
x h y )  =  ( x H y ) )
1715oveqd 5875 . . . . . . . . . . . 12  |-  ( ( ( c  =  C  /\  b  =  B )  /\  h  =  H )  ->  (
z h x )  =  ( z H x ) )
18 simpll 730 . . . . . . . . . . . . . . . 16  |-  ( ( ( c  =  C  /\  b  =  B )  /\  h  =  H )  ->  c  =  C )
1918fveq2d 5529 . . . . . . . . . . . . . . 15  |-  ( ( ( c  =  C  /\  b  =  B )  /\  h  =  H )  ->  (comp `  c )  =  (comp `  C ) )
20 ismon.o . . . . . . . . . . . . . . 15  |-  .x.  =  (comp `  C )
2119, 20syl6eqr 2333 . . . . . . . . . . . . . 14  |-  ( ( ( c  =  C  /\  b  =  B )  /\  h  =  H )  ->  (comp `  c )  =  .x.  )
2221oveqd 5875 . . . . . . . . . . . . 13  |-  ( ( ( c  =  C  /\  b  =  B )  /\  h  =  H )  ->  ( <. z ,  x >. (comp `  c ) y )  =  ( <. z ,  x >.  .x.  y ) )
2322oveqd 5875 . . . . . . . . . . . 12  |-  ( ( ( c  =  C  /\  b  =  B )  /\  h  =  H )  ->  (
f ( <. z ,  x >. (comp `  c
) y ) g )  =  ( f ( <. z ,  x >.  .x.  y ) g ) )
2417, 23mpteq12dv 4098 . . . . . . . . . . 11  |-  ( ( ( c  =  C  /\  b  =  B )  /\  h  =  H )  ->  (
g  e.  ( z h x )  |->  ( f ( <. z ,  x >. (comp `  c
) y ) g ) )  =  ( g  e.  ( z H x )  |->  ( f ( <. z ,  x >.  .x.  y ) g ) ) )
2524cnveqd 4857 . . . . . . . . . 10  |-  ( ( ( c  =  C  /\  b  =  B )  /\  h  =  H )  ->  `' ( g  e.  ( z h x ) 
|->  ( f ( <.
z ,  x >. (comp `  c ) y ) g ) )  =  `' ( g  e.  ( z H x )  |->  ( f (
<. z ,  x >.  .x.  y ) g ) ) )
2625funeqd 5276 . . . . . . . . 9  |-  ( ( ( c  =  C  /\  b  =  B )  /\  h  =  H )  ->  ( Fun  `' ( g  e.  ( z h x )  |->  ( f (
<. z ,  x >. (comp `  c ) y ) g ) )  <->  Fun  `' ( g  e.  ( z H x )  |->  ( f ( <. z ,  x >.  .x.  y ) g ) ) ) )
2714, 26raleqbidv 2748 . . . . . . . 8  |-  ( ( ( c  =  C  /\  b  =  B )  /\  h  =  H )  ->  ( A. z  e.  b  Fun  `' ( g  e.  ( z h x )  |->  ( f (
<. z ,  x >. (comp `  c ) y ) g ) )  <->  A. z  e.  B  Fun  `' ( g  e.  ( z H x )  |->  ( f ( <. z ,  x >.  .x.  y ) g ) ) ) )
2816, 27rabeqbidv 2783 . . . . . . 7  |-  ( ( ( c  =  C  /\  b  =  B )  /\  h  =  H )  ->  { f  e.  ( x h y )  |  A. z  e.  b  Fun  `' ( g  e.  ( z h x ) 
|->  ( f ( <.
z ,  x >. (comp `  c ) y ) g ) ) }  =  { f  e.  ( x H y )  |  A. z  e.  B  Fun  `' ( g  e.  ( z H x )  |->  ( f ( <. z ,  x >.  .x.  y ) g ) ) } )
2914, 14, 28mpt2eq123dv 5910 . . . . . 6  |-  ( ( ( c  =  C  /\  b  =  B )  /\  h  =  H )  ->  (
x  e.  b ,  y  e.  b  |->  { f  e.  ( x h y )  | 
A. z  e.  b  Fun  `' ( g  e.  ( z h x )  |->  ( f ( <. z ,  x >. (comp `  c )
y ) g ) ) } )  =  ( x  e.  B ,  y  e.  B  |->  { f  e.  ( x H y )  |  A. z  e.  B  Fun  `' ( g  e.  ( z H x )  |->  ( f ( <. z ,  x >.  .x.  y ) g ) ) } ) )
309, 13, 29csbied2 3124 . . . . 5  |-  ( ( c  =  C  /\  b  =  B )  ->  [_ (  Hom  `  c
)  /  h ]_ ( x  e.  b ,  y  e.  b  |->  { f  e.  ( x h y )  |  A. z  e.  b  Fun  `' ( g  e.  ( z h x )  |->  ( f ( <. z ,  x >. (comp `  c
) y ) g ) ) } )  =  ( x  e.  B ,  y  e.  B  |->  { f  e.  ( x H y )  |  A. z  e.  B  Fun  `' ( g  e.  ( z H x )  |->  ( f ( <. z ,  x >.  .x.  y ) g ) ) } ) )
314, 7, 30csbied2 3124 . . . 4  |-  ( c  =  C  ->  [_ ( Base `  c )  / 
b ]_ [_ (  Hom  `  c )  /  h ]_ ( x  e.  b ,  y  e.  b 
|->  { f  e.  ( x h y )  |  A. z  e.  b  Fun  `' ( g  e.  ( z h x )  |->  ( f ( <. z ,  x >. (comp `  c
) y ) g ) ) } )  =  ( x  e.  B ,  y  e.  B  |->  { f  e.  ( x H y )  |  A. z  e.  B  Fun  `' ( g  e.  ( z H x )  |->  ( f ( <. z ,  x >.  .x.  y ) g ) ) } ) )
32 df-mon 13633 . . . 4  |- Mono  =  ( c  e.  Cat  |->  [_ ( Base `  c )  /  b ]_ [_ (  Hom  `  c )  /  h ]_ ( x  e.  b ,  y  e.  b  |->  { f  e.  ( x h y )  |  A. z  e.  b  Fun  `' ( g  e.  ( z h x )  |->  ( f ( <. z ,  x >. (comp `  c
) y ) g ) ) } ) )
33 fvex 5539 . . . . . 6  |-  ( Base `  C )  e.  _V
346, 33eqeltri 2353 . . . . 5  |-  B  e. 
_V
3534, 34mpt2ex 6198 . . . 4  |-  ( x  e.  B ,  y  e.  B  |->  { f  e.  ( x H y )  |  A. z  e.  B  Fun  `' ( g  e.  ( z H x ) 
|->  ( f ( <.
z ,  x >.  .x.  y ) g ) ) } )  e. 
_V
3631, 32, 35fvmpt 5602 . . 3  |-  ( C  e.  Cat  ->  (Mono `  C )  =  ( x  e.  B , 
y  e.  B  |->  { f  e.  ( x H y )  | 
A. z  e.  B  Fun  `' ( g  e.  ( z H x )  |->  ( f (
<. z ,  x >.  .x.  y ) g ) ) } ) )
372, 36syl 15 . 2  |-  ( ph  ->  (Mono `  C )  =  ( x  e.  B ,  y  e.  B  |->  { f  e.  ( x H y )  |  A. z  e.  B  Fun  `' ( g  e.  ( z H x )  |->  ( f ( <. z ,  x >.  .x.  y ) g ) ) } ) )
381, 37syl5eq 2327 1  |-  ( ph  ->  M  =  ( x  e.  B ,  y  e.  B  |->  { f  e.  ( x H y )  |  A. z  e.  B  Fun  `' ( g  e.  ( z H x ) 
|->  ( f ( <.
z ,  x >.  .x.  y ) g ) ) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788   [_csb 3081   <.cop 3643    e. cmpt 4077   `'ccnv 4688   Fun wfun 5249   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   Basecbs 13148    Hom chom 13219  compcco 13220   Catccat 13566  Monocmon 13631
This theorem is referenced by:  ismon  13636  monpropd  13640
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-mon 13633
  Copyright terms: Public domain W3C validator