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Theorem moni 13655
Description: Property of a monomorphism. (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 )
moni.z  |-  ( ph  ->  Z  e.  B )
moni.f  |-  ( ph  ->  F  e.  ( X M Y ) )
moni.g  |-  ( ph  ->  G  e.  ( Z H X ) )
moni.k  |-  ( ph  ->  K  e.  ( Z H X ) )
Assertion
Ref Expression
moni  |-  ( ph  ->  ( ( F (
<. Z ,  X >.  .x. 
Y ) G )  =  ( F (
<. Z ,  X >.  .x. 
Y ) K )  <-> 
G  =  K ) )

Proof of Theorem moni
Dummy variables  g  h  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 moni.f . . . . 5  |-  ( ph  ->  F  e.  ( X M Y ) )
2 ismon.b . . . . . 6  |-  B  =  ( Base `  C
)
3 ismon.h . . . . . 6  |-  H  =  (  Hom  `  C
)
4 ismon.o . . . . . 6  |-  .x.  =  (comp `  C )
5 ismon.s . . . . . 6  |-  M  =  (Mono `  C )
6 ismon.c . . . . . 6  |-  ( ph  ->  C  e.  Cat )
7 ismon.x . . . . . 6  |-  ( ph  ->  X  e.  B )
8 ismon.y . . . . . 6  |-  ( ph  ->  Y  e.  B )
92, 3, 4, 5, 6, 7, 8ismon2 13653 . . . . 5  |-  ( ph  ->  ( F  e.  ( X M Y )  <-> 
( F  e.  ( X H Y )  /\  A. z  e.  B  A. g  e.  ( z H X ) A. h  e.  ( z H X ) ( ( F ( <. z ,  X >.  .x.  Y ) g )  =  ( F ( <. z ,  X >.  .x.  Y ) h )  ->  g  =  h ) ) ) )
101, 9mpbid 201 . . . 4  |-  ( ph  ->  ( F  e.  ( X H Y )  /\  A. z  e.  B  A. g  e.  ( z H X ) A. h  e.  ( z H X ) ( ( F ( <. z ,  X >.  .x.  Y ) g )  =  ( F ( <. z ,  X >.  .x.  Y ) h )  ->  g  =  h ) ) )
1110simprd 449 . . 3  |-  ( ph  ->  A. z  e.  B  A. g  e.  (
z H X ) A. h  e.  ( z H X ) ( ( F (
<. z ,  X >.  .x. 
Y ) g )  =  ( F (
<. z ,  X >.  .x. 
Y ) h )  ->  g  =  h ) )
12 moni.z . . . 4  |-  ( ph  ->  Z  e.  B )
13 moni.g . . . . . . 7  |-  ( ph  ->  G  e.  ( Z H X ) )
1413adantr 451 . . . . . 6  |-  ( (
ph  /\  z  =  Z )  ->  G  e.  ( Z H X ) )
15 simpr 447 . . . . . . 7  |-  ( (
ph  /\  z  =  Z )  ->  z  =  Z )
1615oveq1d 5889 . . . . . 6  |-  ( (
ph  /\  z  =  Z )  ->  (
z H X )  =  ( Z H X ) )
1714, 16eleqtrrd 2373 . . . . 5  |-  ( (
ph  /\  z  =  Z )  ->  G  e.  ( z H X ) )
18 moni.k . . . . . . . . 9  |-  ( ph  ->  K  e.  ( Z H X ) )
1918adantr 451 . . . . . . . 8  |-  ( (
ph  /\  z  =  Z )  ->  K  e.  ( Z H X ) )
2019, 16eleqtrrd 2373 . . . . . . 7  |-  ( (
ph  /\  z  =  Z )  ->  K  e.  ( z H X ) )
2120adantr 451 . . . . . 6  |-  ( ( ( ph  /\  z  =  Z )  /\  g  =  G )  ->  K  e.  ( z H X ) )
22 simpllr 735 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  z  =  Z )  /\  g  =  G
)  /\  h  =  K )  ->  z  =  Z )
2322opeq1d 3818 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  z  =  Z )  /\  g  =  G
)  /\  h  =  K )  ->  <. z ,  X >.  =  <. Z ,  X >. )
2423oveq1d 5889 . . . . . . . . 9  |-  ( ( ( ( ph  /\  z  =  Z )  /\  g  =  G
)  /\  h  =  K )  ->  ( <. z ,  X >.  .x. 
Y )  =  (
<. Z ,  X >.  .x. 
Y ) )
25 eqidd 2297 . . . . . . . . 9  |-  ( ( ( ( ph  /\  z  =  Z )  /\  g  =  G
)  /\  h  =  K )  ->  F  =  F )
26 simplr 731 . . . . . . . . 9  |-  ( ( ( ( ph  /\  z  =  Z )  /\  g  =  G
)  /\  h  =  K )  ->  g  =  G )
2724, 25, 26oveq123d 5895 . . . . . . . 8  |-  ( ( ( ( ph  /\  z  =  Z )  /\  g  =  G
)  /\  h  =  K )  ->  ( F ( <. z ,  X >.  .x.  Y ) g )  =  ( F ( <. Z ,  X >.  .x.  Y ) G ) )
28 simpr 447 . . . . . . . . 9  |-  ( ( ( ( ph  /\  z  =  Z )  /\  g  =  G
)  /\  h  =  K )  ->  h  =  K )
2924, 25, 28oveq123d 5895 . . . . . . . 8  |-  ( ( ( ( ph  /\  z  =  Z )  /\  g  =  G
)  /\  h  =  K )  ->  ( F ( <. z ,  X >.  .x.  Y ) h )  =  ( F ( <. Z ,  X >.  .x.  Y ) K ) )
3027, 29eqeq12d 2310 . . . . . . 7  |-  ( ( ( ( ph  /\  z  =  Z )  /\  g  =  G
)  /\  h  =  K )  ->  (
( F ( <.
z ,  X >.  .x. 
Y ) g )  =  ( F (
<. z ,  X >.  .x. 
Y ) h )  <-> 
( F ( <. Z ,  X >.  .x. 
Y ) G )  =  ( F (
<. Z ,  X >.  .x. 
Y ) K ) ) )
3126, 28eqeq12d 2310 . . . . . . 7  |-  ( ( ( ( ph  /\  z  =  Z )  /\  g  =  G
)  /\  h  =  K )  ->  (
g  =  h  <->  G  =  K ) )
3230, 31imbi12d 311 . . . . . 6  |-  ( ( ( ( ph  /\  z  =  Z )  /\  g  =  G
)  /\  h  =  K )  ->  (
( ( F (
<. z ,  X >.  .x. 
Y ) g )  =  ( F (
<. z ,  X >.  .x. 
Y ) h )  ->  g  =  h )  <->  ( ( F ( <. Z ,  X >.  .x.  Y ) G )  =  ( F ( <. Z ,  X >.  .x.  Y ) K )  ->  G  =  K ) ) )
3321, 32rspcdv 2900 . . . . 5  |-  ( ( ( ph  /\  z  =  Z )  /\  g  =  G )  ->  ( A. h  e.  (
z H X ) ( ( F (
<. z ,  X >.  .x. 
Y ) g )  =  ( F (
<. z ,  X >.  .x. 
Y ) h )  ->  g  =  h )  ->  ( ( F ( <. Z ,  X >.  .x.  Y ) G )  =  ( F ( <. Z ,  X >.  .x.  Y ) K )  ->  G  =  K ) ) )
3417, 33rspcimdv 2898 . . . 4  |-  ( (
ph  /\  z  =  Z )  ->  ( A. g  e.  (
z H X ) A. h  e.  ( z H X ) ( ( F (
<. z ,  X >.  .x. 
Y ) g )  =  ( F (
<. z ,  X >.  .x. 
Y ) h )  ->  g  =  h )  ->  ( ( F ( <. Z ,  X >.  .x.  Y ) G )  =  ( F ( <. Z ,  X >.  .x.  Y ) K )  ->  G  =  K ) ) )
3512, 34rspcimdv 2898 . . 3  |-  ( ph  ->  ( A. z  e.  B  A. g  e.  ( z H X ) A. h  e.  ( z H X ) ( ( F ( <. z ,  X >.  .x.  Y ) g )  =  ( F ( <. z ,  X >.  .x.  Y ) h )  ->  g  =  h )  ->  (
( F ( <. Z ,  X >.  .x. 
Y ) G )  =  ( F (
<. Z ,  X >.  .x. 
Y ) K )  ->  G  =  K ) ) )
3611, 35mpd 14 . 2  |-  ( ph  ->  ( ( F (
<. Z ,  X >.  .x. 
Y ) G )  =  ( F (
<. Z ,  X >.  .x. 
Y ) K )  ->  G  =  K ) )
37 oveq2 5882 . 2  |-  ( G  =  K  ->  ( F ( <. Z ,  X >.  .x.  Y ) G )  =  ( F ( <. Z ,  X >.  .x.  Y ) K ) )
3836, 37impbid1 194 1  |-  ( ph  ->  ( ( F (
<. Z ,  X >.  .x. 
Y ) G )  =  ( F (
<. Z ,  X >.  .x. 
Y ) K )  <-> 
G  =  K ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   <.cop 3656   ` cfv 5271  (class class class)co 5874   Basecbs 13164    Hom chom 13235  compcco 13236   Catccat 13582  Monocmon 13647
This theorem is referenced by:  epii  13662  monsect  13697  fthmon  13817  setcmon  13935
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-cat 13586  df-mon 13649
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