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Theorem moni 13954
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 13952 . . . . 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 202 . . . 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 450 . . 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 452 . . . . . 6  |-  ( (
ph  /\  z  =  Z )  ->  G  e.  ( Z H X ) )
15 simpr 448 . . . . . . 7  |-  ( (
ph  /\  z  =  Z )  ->  z  =  Z )
1615oveq1d 6088 . . . . . 6  |-  ( (
ph  /\  z  =  Z )  ->  (
z H X )  =  ( Z H X ) )
1714, 16eleqtrrd 2512 . . . . 5  |-  ( (
ph  /\  z  =  Z )  ->  G  e.  ( z H X ) )
18 moni.k . . . . . . . . 9  |-  ( ph  ->  K  e.  ( Z H X ) )
1918adantr 452 . . . . . . . 8  |-  ( (
ph  /\  z  =  Z )  ->  K  e.  ( Z H X ) )
2019, 16eleqtrrd 2512 . . . . . . 7  |-  ( (
ph  /\  z  =  Z )  ->  K  e.  ( z H X ) )
2120adantr 452 . . . . . 6  |-  ( ( ( ph  /\  z  =  Z )  /\  g  =  G )  ->  K  e.  ( z H X ) )
22 simpllr 736 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  z  =  Z )  /\  g  =  G
)  /\  h  =  K )  ->  z  =  Z )
2322opeq1d 3982 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  z  =  Z )  /\  g  =  G
)  /\  h  =  K )  ->  <. z ,  X >.  =  <. Z ,  X >. )
2423oveq1d 6088 . . . . . . . . 9  |-  ( ( ( ( ph  /\  z  =  Z )  /\  g  =  G
)  /\  h  =  K )  ->  ( <. z ,  X >.  .x. 
Y )  =  (
<. Z ,  X >.  .x. 
Y ) )
25 eqidd 2436 . . . . . . . . 9  |-  ( ( ( ( ph  /\  z  =  Z )  /\  g  =  G
)  /\  h  =  K )  ->  F  =  F )
26 simplr 732 . . . . . . . . 9  |-  ( ( ( ( ph  /\  z  =  Z )  /\  g  =  G
)  /\  h  =  K )  ->  g  =  G )
2724, 25, 26oveq123d 6094 . . . . . . . 8  |-  ( ( ( ( ph  /\  z  =  Z )  /\  g  =  G
)  /\  h  =  K )  ->  ( F ( <. z ,  X >.  .x.  Y ) g )  =  ( F ( <. Z ,  X >.  .x.  Y ) G ) )
28 simpr 448 . . . . . . . . 9  |-  ( ( ( ( ph  /\  z  =  Z )  /\  g  =  G
)  /\  h  =  K )  ->  h  =  K )
2924, 25, 28oveq123d 6094 . . . . . . . 8  |-  ( ( ( ( ph  /\  z  =  Z )  /\  g  =  G
)  /\  h  =  K )  ->  ( F ( <. z ,  X >.  .x.  Y ) h )  =  ( F ( <. Z ,  X >.  .x.  Y ) K ) )
3027, 29eqeq12d 2449 . . . . . . 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 2449 . . . . . . 7  |-  ( ( ( ( ph  /\  z  =  Z )  /\  g  =  G
)  /\  h  =  K )  ->  (
g  =  h  <->  G  =  K ) )
3230, 31imbi12d 312 . . . . . 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 3047 . . . . 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 3045 . . . 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 3045 . . 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 6081 . 2  |-  ( G  =  K  ->  ( F ( <. Z ,  X >.  .x.  Y ) G )  =  ( F ( <. Z ,  X >.  .x.  Y ) K ) )
3836, 37impbid1 195 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   <.cop 3809   ` cfv 5446  (class class class)co 6073   Basecbs 13461    Hom chom 13532  compcco 13533   Catccat 13881  Monocmon 13946
This theorem is referenced by:  epii  13961  monsect  13996  fthmon  14116  setcmon  14234
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-cat 13885  df-mon 13948
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