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Theorem ismon 13652
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 13651 . . . 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 5892 . . . . 5  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( x H y )  =  ( X H Y ) )
10 eqidd 2297 . . . . . 6  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  B  =  B )
117oveq2d 5890 . . . . . . . . 9  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( z H x )  =  ( z H X ) )
127opeq2d 3819 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  <. z ,  x >.  = 
<. z ,  X >. )
1312, 8oveq12d 5892 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( <. z ,  x >.  .x.  y )  =  ( <. z ,  X >.  .x.  Y ) )
1413oveqd 5891 . . . . . . . . 9  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( f ( <.
z ,  x >.  .x.  y ) g )  =  ( f (
<. z ,  X >.  .x. 
Y ) g ) )
1511, 14mpteq12dv 4114 . . . . . . . 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 4873 . . . . . . 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 5292 . . . . . 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 2761 . . . . 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 2796 . . . 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 5899 . . . . . 6  |-  ( X H Y )  e. 
_V
2322rabex 4181 . . . . 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 5991 . . 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 2363 . 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 2297 . . . 4  |-  ( f  =  F  ->  B  =  B )
28 eqidd 2297 . . . . . . 7  |-  ( f  =  F  ->  (
z H X )  =  ( z H X ) )
29 oveq1 5881 . . . . . . 7  |-  ( f  =  F  ->  (
f ( <. z ,  X >.  .x.  Y ) g )  =  ( F ( <. z ,  X >.  .x.  Y ) g ) )
3028, 29mpteq12dv 4114 . . . . . 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 4873 . . . . 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 5292 . . . 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 2761 . . 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 2936 . 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 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801   <.cop 3656    e. cmpt 4093   `'ccnv 4704   Fun wfun 5265   ` cfv 5271  (class class class)co 5874   Basecbs 13164    Hom chom 13235  compcco 13236   Catccat 13582  Monocmon 13647
This theorem is referenced by:  ismon2  13653  monhom  13654  isepi  13659
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-mon 13649
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