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Theorem ismon 13636
Description: Definition of a monomorphism in a category. (Contributed by Mario Carneiro, 2-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 )
ismon.x  |-  ( ph  ->  X  e.  B )
ismon.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
ismon  |-  ( ph  ->  ( F  e.  ( X M Y )  <-> 
( F  e.  ( X H Y )  /\  A. z  e.  B  Fun  `' ( g  e.  ( z H X )  |->  ( F ( <. z ,  X >.  .x.  Y ) g ) ) ) ) )
Distinct variable groups:    z, g, B    ph, g, z    C, g, z    g, H, z    .x. , g, z    g, F, z    g, X, z   
g, Y, z
Allowed substitution hints:    M( z, g)

Proof of Theorem ismon
Dummy variables  f  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ismon.b . . . . 5  |-  B  =  ( Base `  C
)
2 ismon.h . . . . 5  |-  H  =  (  Hom  `  C
)
3 ismon.o . . . . 5  |-  .x.  =  (comp `  C )
4 ismon.s . . . . 5  |-  M  =  (Mono `  C )
5 ismon.c . . . . 5  |-  ( ph  ->  C  e.  Cat )
61, 2, 3, 4, 5monfval 13635 . . . 4  |-  ( 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 ) ) } ) )
7 simprl 732 . . . . . 6  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  x  =  X )
8 simprr 733 . . . . . 6  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
y  =  Y )
97, 8oveq12d 5876 . . . . 5  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( x H y )  =  ( X H Y ) )
10 eqidd 2284 . . . . . 6  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  B  =  B )
117oveq2d 5874 . . . . . . . . 9  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( z H x )  =  ( z H X ) )
127opeq2d 3803 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  <. z ,  x >.  = 
<. z ,  X >. )
1312, 8oveq12d 5876 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( <. z ,  x >.  .x.  y )  =  ( <. z ,  X >.  .x.  Y ) )
1413oveqd 5875 . . . . . . . . 9  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( f ( <.
z ,  x >.  .x.  y ) g )  =  ( f (
<. z ,  X >.  .x. 
Y ) g ) )
1511, 14mpteq12dv 4098 . . . . . . . 8  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( g  e.  ( z H x ) 
|->  ( f ( <.
z ,  x >.  .x.  y ) g ) )  =  ( g  e.  ( z H X )  |->  ( f ( <. z ,  X >.  .x.  Y ) g ) ) )
1615cnveqd 4857 . . . . . . 7  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  `' ( g  e.  ( z H x )  |->  ( f (
<. z ,  x >.  .x.  y ) g ) )  =  `' ( g  e.  ( z H X )  |->  ( f ( <. z ,  X >.  .x.  Y ) g ) ) )
1716funeqd 5276 . . . . . 6  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( Fun  `' (
g  e.  ( z H x )  |->  ( f ( <. z ,  x >.  .x.  y ) g ) )  <->  Fun  `' ( g  e.  ( z H X )  |->  ( f ( <. z ,  X >.  .x.  Y ) g ) ) ) )
1810, 17raleqbidv 2748 . . . . 5  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( A. z  e.  B  Fun  `' ( g  e.  ( z H x )  |->  ( f ( <. z ,  x >.  .x.  y ) g ) )  <->  A. z  e.  B  Fun  `' ( g  e.  ( z H X )  |->  ( f ( <. z ,  X >.  .x.  Y ) g ) ) ) )
199, 18rabeqbidv 2783 . . . 4  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  { f  e.  ( x H y )  |  A. z  e.  B  Fun  `' ( g  e.  ( z H x )  |->  ( f ( <. z ,  x >.  .x.  y ) g ) ) }  =  { f  e.  ( X H Y )  |  A. z  e.  B  Fun  `' ( g  e.  ( z H X )  |->  ( f ( <. z ,  X >.  .x.  Y ) g ) ) } )
20 ismon.x . . . 4  |-  ( ph  ->  X  e.  B )
21 ismon.y . . . 4  |-  ( ph  ->  Y  e.  B )
22 ovex 5883 . . . . . 6  |-  ( X H Y )  e. 
_V
2322rabex 4165 . . . . 5  |-  { f  e.  ( X H Y )  |  A. z  e.  B  Fun  `' ( g  e.  ( z H X ) 
|->  ( f ( <.
z ,  X >.  .x. 
Y ) g ) ) }  e.  _V
2423a1i 10 . . . 4  |-  ( ph  ->  { f  e.  ( X H Y )  |  A. z  e.  B  Fun  `' ( g  e.  ( z H X )  |->  ( f ( <. z ,  X >.  .x.  Y ) g ) ) }  e.  _V )
256, 19, 20, 21, 24ovmpt2d 5975 . . 3  |-  ( ph  ->  ( X M Y )  =  { f  e.  ( X H Y )  |  A. z  e.  B  Fun  `' ( g  e.  ( z H X ) 
|->  ( f ( <.
z ,  X >.  .x. 
Y ) g ) ) } )
2625eleq2d 2350 . 2  |-  ( ph  ->  ( F  e.  ( X M Y )  <-> 
F  e.  { f  e.  ( X H Y )  |  A. z  e.  B  Fun  `' ( g  e.  ( z H X ) 
|->  ( f ( <.
z ,  X >.  .x. 
Y ) g ) ) } ) )
27 eqidd 2284 . . . 4  |-  ( f  =  F  ->  B  =  B )
28 eqidd 2284 . . . . . . 7  |-  ( f  =  F  ->  (
z H X )  =  ( z H X ) )
29 oveq1 5865 . . . . . . 7  |-  ( f  =  F  ->  (
f ( <. z ,  X >.  .x.  Y ) g )  =  ( F ( <. z ,  X >.  .x.  Y ) g ) )
3028, 29mpteq12dv 4098 . . . . . 6  |-  ( f  =  F  ->  (
g  e.  ( z H X )  |->  ( f ( <. z ,  X >.  .x.  Y ) g ) )  =  ( g  e.  ( z H X ) 
|->  ( F ( <.
z ,  X >.  .x. 
Y ) g ) ) )
3130cnveqd 4857 . . . . 5  |-  ( f  =  F  ->  `' ( g  e.  ( z H X ) 
|->  ( f ( <.
z ,  X >.  .x. 
Y ) g ) )  =  `' ( g  e.  ( z H X )  |->  ( F ( <. z ,  X >.  .x.  Y ) g ) ) )
3231funeqd 5276 . . . 4  |-  ( f  =  F  ->  ( Fun  `' ( g  e.  ( z H X )  |->  ( f (
<. z ,  X >.  .x. 
Y ) g ) )  <->  Fun  `' ( g  e.  ( z H X )  |->  ( F ( <. z ,  X >.  .x.  Y ) g ) ) ) )
3327, 32raleqbidv 2748 . . 3  |-  ( f  =  F  ->  ( A. z  e.  B  Fun  `' ( g  e.  ( z H X )  |->  ( f (
<. z ,  X >.  .x. 
Y ) g ) )  <->  A. z  e.  B  Fun  `' ( g  e.  ( z H X )  |->  ( F (
<. z ,  X >.  .x. 
Y ) g ) ) ) )
3433elrab 2923 . 2  |-  ( F  e.  { f  e.  ( X H Y )  |  A. z  e.  B  Fun  `' ( g  e.  ( z H X )  |->  ( f ( <. z ,  X >.  .x.  Y ) g ) ) }  <-> 
( F  e.  ( X H Y )  /\  A. z  e.  B  Fun  `' ( g  e.  ( z H X )  |->  ( F ( <. z ,  X >.  .x.  Y ) g ) ) ) )
3526, 34syl6bb 252 1  |-  ( ph  ->  ( F  e.  ( X M Y )  <-> 
( 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    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788   <.cop 3643    e. cmpt 4077   `'ccnv 4688   Fun wfun 5249   ` cfv 5255  (class class class)co 5858   Basecbs 13148    Hom chom 13219  compcco 13220   Catccat 13566  Monocmon 13631
This theorem is referenced by:  ismon2  13637  monhom  13638  isepi  13643
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
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