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Theorem moni 15009
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 15007 . . . . 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 210 . . . 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 463 . . 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 465 . . . . . 6  |-  ( (
ph  /\  z  =  Z )  ->  G  e.  ( Z H X ) )
15 simpr 461 . . . . . . 7  |-  ( (
ph  /\  z  =  Z )  ->  z  =  Z )
1615oveq1d 6310 . . . . . 6  |-  ( (
ph  /\  z  =  Z )  ->  (
z H X )  =  ( Z H X ) )
1714, 16eleqtrrd 2558 . . . . 5  |-  ( (
ph  /\  z  =  Z )  ->  G  e.  ( z H X ) )
18 moni.k . . . . . . . . 9  |-  ( ph  ->  K  e.  ( Z H X ) )
1918adantr 465 . . . . . . . 8  |-  ( (
ph  /\  z  =  Z )  ->  K  e.  ( Z H X ) )
2019, 16eleqtrrd 2558 . . . . . . 7  |-  ( (
ph  /\  z  =  Z )  ->  K  e.  ( z H X ) )
2120adantr 465 . . . . . 6  |-  ( ( ( ph  /\  z  =  Z )  /\  g  =  G )  ->  K  e.  ( z H X ) )
22 simpllr 758 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  z  =  Z )  /\  g  =  G
)  /\  h  =  K )  ->  z  =  Z )
2322opeq1d 4225 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  z  =  Z )  /\  g  =  G
)  /\  h  =  K )  ->  <. z ,  X >.  =  <. Z ,  X >. )
2423oveq1d 6310 . . . . . . . . 9  |-  ( ( ( ( ph  /\  z  =  Z )  /\  g  =  G
)  /\  h  =  K )  ->  ( <. z ,  X >.  .x. 
Y )  =  (
<. Z ,  X >.  .x. 
Y ) )
25 eqidd 2468 . . . . . . . . 9  |-  ( ( ( ( ph  /\  z  =  Z )  /\  g  =  G
)  /\  h  =  K )  ->  F  =  F )
26 simplr 754 . . . . . . . . 9  |-  ( ( ( ( ph  /\  z  =  Z )  /\  g  =  G
)  /\  h  =  K )  ->  g  =  G )
2724, 25, 26oveq123d 6316 . . . . . . . 8  |-  ( ( ( ( ph  /\  z  =  Z )  /\  g  =  G
)  /\  h  =  K )  ->  ( F ( <. z ,  X >.  .x.  Y ) g )  =  ( F ( <. Z ,  X >.  .x.  Y ) G ) )
28 simpr 461 . . . . . . . . 9  |-  ( ( ( ( ph  /\  z  =  Z )  /\  g  =  G
)  /\  h  =  K )  ->  h  =  K )
2924, 25, 28oveq123d 6316 . . . . . . . 8  |-  ( ( ( ( ph  /\  z  =  Z )  /\  g  =  G
)  /\  h  =  K )  ->  ( F ( <. z ,  X >.  .x.  Y ) h )  =  ( F ( <. Z ,  X >.  .x.  Y ) K ) )
3027, 29eqeq12d 2489 . . . . . . 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 2489 . . . . . . 7  |-  ( ( ( ( ph  /\  z  =  Z )  /\  g  =  G
)  /\  h  =  K )  ->  (
g  =  h  <->  G  =  K ) )
3230, 31imbi12d 320 . . . . . 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 3222 . . . . 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 3220 . . . 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 3220 . . 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 15 . 2  |-  ( ph  ->  ( ( F (
<. Z ,  X >.  .x. 
Y ) G )  =  ( F (
<. Z ,  X >.  .x. 
Y ) K )  ->  G  =  K ) )
37 oveq2 6303 . 2  |-  ( G  =  K  ->  ( F ( <. Z ,  X >.  .x.  Y ) G )  =  ( F ( <. Z ,  X >.  .x.  Y ) K ) )
3836, 37impbid1 203 1  |-  ( ph  ->  ( ( F (
<. Z ,  X >.  .x. 
Y ) G )  =  ( F (
<. Z ,  X >.  .x. 
Y ) K )  <-> 
G  =  K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   <.cop 4039   ` cfv 5594  (class class class)co 6295   Basecbs 14507   Hom chom 14583  compcco 14584   Catccat 14936  Monocmon 15001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-cat 14940  df-mon 15003
This theorem is referenced by:  epii  15016  monsect  15051  fthmon  15171  setcmon  15289
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