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Theorem ismon 13959
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 13958 . . . 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 733 . . . . . 6  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  x  =  X )
8 simprr 734 . . . . . 6  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
y  =  Y )
97, 8oveq12d 6099 . . . . 5  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( x H y )  =  ( X H Y ) )
107oveq2d 6097 . . . . . . . . 9  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( z H x )  =  ( z H X ) )
117opeq2d 3991 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  ->  <. z ,  x >.  = 
<. z ,  X >. )
1211, 8oveq12d 6099 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( <. z ,  x >.  .x.  y )  =  ( <. z ,  X >.  .x.  Y ) )
1312oveqd 6098 . . . . . . . . 9  |-  ( (
ph  /\  ( x  =  X  /\  y  =  Y ) )  -> 
( f ( <.
z ,  x >.  .x.  y ) g )  =  ( f (
<. z ,  X >.  .x. 
Y ) g ) )
1410, 13mpteq12dv 4287 . . . . . . . 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 ) ) )
1514cnveqd 5048 . . . . . . 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 ) ) )
1615funeqd 5475 . . . . . 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 ) ) ) )
1716ralbidv 2725 . . . . 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 ) ) ) )
189, 17rabeqbidv 2951 . . . 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 ) ) } )
19 ismon.x . . . 4  |-  ( ph  ->  X  e.  B )
20 ismon.y . . . 4  |-  ( ph  ->  Y  e.  B )
21 ovex 6106 . . . . . 6  |-  ( X H Y )  e. 
_V
2221rabex 4354 . . . . 5  |-  { f  e.  ( X H Y )  |  A. z  e.  B  Fun  `' ( g  e.  ( z H X ) 
|->  ( f ( <.
z ,  X >.  .x. 
Y ) g ) ) }  e.  _V
2322a1i 11 . . . 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 )
246, 18, 19, 20, 23ovmpt2d 6201 . . 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 ) ) } )
2524eleq2d 2503 . 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 ) ) } ) )
26 oveq1 6088 . . . . . . 7  |-  ( f  =  F  ->  (
f ( <. z ,  X >.  .x.  Y ) g )  =  ( F ( <. z ,  X >.  .x.  Y ) g ) )
2726mpteq2dv 4296 . . . . . 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 ) ) )
2827cnveqd 5048 . . . . 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 ) ) )
2928funeqd 5475 . . . 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 ) ) ) )
3029ralbidv 2725 . . 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 ) ) ) )
3130elrab 3092 . 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 ) ) ) )
3225, 31syl6bb 253 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   {crab 2709   _Vcvv 2956   <.cop 3817    e. cmpt 4266   `'ccnv 4877   Fun wfun 5448   ` cfv 5454  (class class class)co 6081   Basecbs 13469    Hom chom 13540  compcco 13541   Catccat 13889  Monocmon 13954
This theorem is referenced by:  ismon2  13960  monhom  13961  isepi  13966
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-mon 13956
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