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Theorem monpropd 13963
Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same monomorphisms. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
monpropd.3  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  D ) )
monpropd.4  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
monpropd.c  |-  ( ph  ->  C  e.  Cat )
monpropd.d  |-  ( ph  ->  D  e.  Cat )
Assertion
Ref Expression
monpropd  |-  ( ph  ->  (Mono `  C )  =  (Mono `  D )
)

Proof of Theorem monpropd
Dummy variables  a 
b  c  f  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2436 . . . . . . . . . . . 12  |-  ( Base `  C )  =  (
Base `  C )
2 eqid 2436 . . . . . . . . . . . 12  |-  (  Hom  `  C )  =  (  Hom  `  C )
3 eqid 2436 . . . . . . . . . . . 12  |-  (  Hom  `  D )  =  (  Hom  `  D )
4 monpropd.3 . . . . . . . . . . . . . 14  |-  ( ph  ->  (  Homf 
`  C )  =  (  Homf 
`  D ) )
54ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  a  e.  ( Base `  C
) )  /\  b  e.  ( Base `  C
) )  ->  (  Homf  `  C )  =  (  Homf 
`  D ) )
65ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  a  e.  ( Base `  C ) )  /\  b  e.  ( Base `  C ) )  /\  f  e.  ( a
(  Hom  `  C ) b ) )  /\  c  e.  ( Base `  C ) )  -> 
(  Homf 
`  C )  =  (  Homf 
`  D ) )
7 simpr 448 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  a  e.  ( Base `  C ) )  /\  b  e.  ( Base `  C ) )  /\  f  e.  ( a
(  Hom  `  C ) b ) )  /\  c  e.  ( Base `  C ) )  -> 
c  e.  ( Base `  C ) )
8 simp-4r 744 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  a  e.  ( Base `  C ) )  /\  b  e.  ( Base `  C ) )  /\  f  e.  ( a
(  Hom  `  C ) b ) )  /\  c  e.  ( Base `  C ) )  -> 
a  e.  ( Base `  C ) )
91, 2, 3, 6, 7, 8homfeqval 13923 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  a  e.  ( Base `  C ) )  /\  b  e.  ( Base `  C ) )  /\  f  e.  ( a
(  Hom  `  C ) b ) )  /\  c  e.  ( Base `  C ) )  -> 
( c (  Hom  `  C ) a )  =  ( c (  Hom  `  D )
a ) )
10 eqid 2436 . . . . . . . . . . . 12  |-  (comp `  C )  =  (comp `  C )
11 eqid 2436 . . . . . . . . . . . 12  |-  (comp `  D )  =  (comp `  D )
124ad5antr 715 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  a  e.  ( Base `  C )
)  /\  b  e.  ( Base `  C )
)  /\  f  e.  ( a (  Hom  `  C ) b ) )  /\  c  e.  ( Base `  C
) )  /\  g  e.  ( c (  Hom  `  C ) a ) )  ->  (  Homf  `  C
)  =  (  Homf  `  D ) )
13 monpropd.4 . . . . . . . . . . . . 13  |-  ( ph  ->  (compf `  C )  =  (compf `  D ) )
1413ad5antr 715 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  a  e.  ( Base `  C )
)  /\  b  e.  ( Base `  C )
)  /\  f  e.  ( a (  Hom  `  C ) b ) )  /\  c  e.  ( Base `  C
) )  /\  g  e.  ( c (  Hom  `  C ) a ) )  ->  (compf `  C )  =  (compf `  D ) )
15 simplr 732 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  a  e.  ( Base `  C )
)  /\  b  e.  ( Base `  C )
)  /\  f  e.  ( a (  Hom  `  C ) b ) )  /\  c  e.  ( Base `  C
) )  /\  g  e.  ( c (  Hom  `  C ) a ) )  ->  c  e.  ( Base `  C )
)
16 simp-5r 746 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  a  e.  ( Base `  C )
)  /\  b  e.  ( Base `  C )
)  /\  f  e.  ( a (  Hom  `  C ) b ) )  /\  c  e.  ( Base `  C
) )  /\  g  e.  ( c (  Hom  `  C ) a ) )  ->  a  e.  ( Base `  C )
)
17 simp-4r 744 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  a  e.  ( Base `  C )
)  /\  b  e.  ( Base `  C )
)  /\  f  e.  ( a (  Hom  `  C ) b ) )  /\  c  e.  ( Base `  C
) )  /\  g  e.  ( c (  Hom  `  C ) a ) )  ->  b  e.  ( Base `  C )
)
18 simpr 448 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  a  e.  ( Base `  C )
)  /\  b  e.  ( Base `  C )
)  /\  f  e.  ( a (  Hom  `  C ) b ) )  /\  c  e.  ( Base `  C
) )  /\  g  e.  ( c (  Hom  `  C ) a ) )  ->  g  e.  ( c (  Hom  `  C ) a ) )
19 simpllr 736 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  a  e.  ( Base `  C )
)  /\  b  e.  ( Base `  C )
)  /\  f  e.  ( a (  Hom  `  C ) b ) )  /\  c  e.  ( Base `  C
) )  /\  g  e.  ( c (  Hom  `  C ) a ) )  ->  f  e.  ( a (  Hom  `  C ) b ) )
201, 2, 10, 11, 12, 14, 15, 16, 17, 18, 19comfeqval 13934 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  a  e.  ( Base `  C )
)  /\  b  e.  ( Base `  C )
)  /\  f  e.  ( a (  Hom  `  C ) b ) )  /\  c  e.  ( Base `  C
) )  /\  g  e.  ( c (  Hom  `  C ) a ) )  ->  ( f
( <. c ,  a
>. (comp `  C )
b ) g )  =  ( f (
<. c ,  a >.
(comp `  D )
b ) g ) )
219, 20mpteq12dva 4286 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  a  e.  ( Base `  C ) )  /\  b  e.  ( Base `  C ) )  /\  f  e.  ( a
(  Hom  `  C ) b ) )  /\  c  e.  ( Base `  C ) )  -> 
( g  e.  ( c (  Hom  `  C
) a )  |->  ( f ( <. c ,  a >. (comp `  C ) b ) g ) )  =  ( g  e.  ( c (  Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) )
2221cnveqd 5048 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  a  e.  ( Base `  C ) )  /\  b  e.  ( Base `  C ) )  /\  f  e.  ( a
(  Hom  `  C ) b ) )  /\  c  e.  ( Base `  C ) )  ->  `' ( g  e.  ( c (  Hom  `  C ) a ) 
|->  ( f ( <.
c ,  a >.
(comp `  C )
b ) g ) )  =  `' ( g  e.  ( c (  Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) )
2322funeqd 5475 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  a  e.  ( Base `  C ) )  /\  b  e.  ( Base `  C ) )  /\  f  e.  ( a
(  Hom  `  C ) b ) )  /\  c  e.  ( Base `  C ) )  -> 
( Fun  `' (
g  e.  ( c (  Hom  `  C
) a )  |->  ( f ( <. c ,  a >. (comp `  C ) b ) g ) )  <->  Fun  `' ( g  e.  ( c (  Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) ) )
2423ralbidva 2721 . . . . . . 7  |-  ( ( ( ( ph  /\  a  e.  ( Base `  C ) )  /\  b  e.  ( Base `  C ) )  /\  f  e.  ( a
(  Hom  `  C ) b ) )  -> 
( A. c  e.  ( Base `  C
) Fun  `' (
g  e.  ( c (  Hom  `  C
) a )  |->  ( f ( <. c ,  a >. (comp `  C ) b ) g ) )  <->  A. c  e.  ( Base `  C
) Fun  `' (
g  e.  ( c (  Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) ) )
2524rabbidva 2947 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ( Base `  C
) )  /\  b  e.  ( Base `  C
) )  ->  { f  e.  ( a (  Hom  `  C )
b )  |  A. c  e.  ( Base `  C ) Fun  `' ( g  e.  ( c (  Hom  `  C
) a )  |->  ( f ( <. c ,  a >. (comp `  C ) b ) g ) ) }  =  { f  e.  ( a (  Hom  `  C ) b )  |  A. c  e.  ( Base `  C
) Fun  `' (
g  e.  ( c (  Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) } )
26 simplr 732 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ( Base `  C
) )  /\  b  e.  ( Base `  C
) )  ->  a  e.  ( Base `  C
) )
27 simpr 448 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ( Base `  C
) )  /\  b  e.  ( Base `  C
) )  ->  b  e.  ( Base `  C
) )
281, 2, 3, 5, 26, 27homfeqval 13923 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ( Base `  C
) )  /\  b  e.  ( Base `  C
) )  ->  (
a (  Hom  `  C
) b )  =  ( a (  Hom  `  D ) b ) )
294homfeqbas 13922 . . . . . . . . 9  |-  ( ph  ->  ( Base `  C
)  =  ( Base `  D ) )
3029ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  ( Base `  C
) )  /\  b  e.  ( Base `  C
) )  ->  ( Base `  C )  =  ( Base `  D
) )
3130raleqdv 2910 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  ( Base `  C
) )  /\  b  e.  ( Base `  C
) )  ->  ( A. c  e.  ( Base `  C ) Fun  `' ( g  e.  ( c (  Hom  `  D ) a ) 
|->  ( f ( <.
c ,  a >.
(comp `  D )
b ) g ) )  <->  A. c  e.  (
Base `  D ) Fun  `' ( g  e.  ( c (  Hom  `  D ) a ) 
|->  ( f ( <.
c ,  a >.
(comp `  D )
b ) g ) ) ) )
3228, 31rabeqbidv 2951 . . . . . 6  |-  ( ( ( ph  /\  a  e.  ( Base `  C
) )  /\  b  e.  ( Base `  C
) )  ->  { f  e.  ( a (  Hom  `  C )
b )  |  A. c  e.  ( Base `  C ) Fun  `' ( g  e.  ( c (  Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) }  =  { f  e.  ( a (  Hom  `  D ) b )  |  A. c  e.  ( Base `  D
) Fun  `' (
g  e.  ( c (  Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) } )
3325, 32eqtrd 2468 . . . . 5  |-  ( ( ( ph  /\  a  e.  ( Base `  C
) )  /\  b  e.  ( Base `  C
) )  ->  { f  e.  ( a (  Hom  `  C )
b )  |  A. c  e.  ( Base `  C ) Fun  `' ( g  e.  ( c (  Hom  `  C
) a )  |->  ( f ( <. c ,  a >. (comp `  C ) b ) g ) ) }  =  { f  e.  ( a (  Hom  `  D ) b )  |  A. c  e.  ( Base `  D
) Fun  `' (
g  e.  ( c (  Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) } )
34333impa 1148 . . . 4  |-  ( (
ph  /\  a  e.  ( Base `  C )  /\  b  e.  ( Base `  C ) )  ->  { f  e.  ( a (  Hom  `  C ) b )  |  A. c  e.  ( Base `  C
) Fun  `' (
g  e.  ( c (  Hom  `  C
) a )  |->  ( f ( <. c ,  a >. (comp `  C ) b ) g ) ) }  =  { f  e.  ( a (  Hom  `  D ) b )  |  A. c  e.  ( Base `  D
) Fun  `' (
g  e.  ( c (  Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) } )
3534mpt2eq3dva 6138 . . 3  |-  ( ph  ->  ( a  e.  (
Base `  C ) ,  b  e.  ( Base `  C )  |->  { f  e.  ( a (  Hom  `  C
) b )  | 
A. c  e.  (
Base `  C ) Fun  `' ( g  e.  ( c (  Hom  `  C ) a ) 
|->  ( f ( <.
c ,  a >.
(comp `  C )
b ) g ) ) } )  =  ( a  e.  (
Base `  C ) ,  b  e.  ( Base `  C )  |->  { f  e.  ( a (  Hom  `  D
) b )  | 
A. c  e.  (
Base `  D ) Fun  `' ( g  e.  ( c (  Hom  `  D ) a ) 
|->  ( f ( <.
c ,  a >.
(comp `  D )
b ) g ) ) } ) )
36 mpt2eq12 6134 . . . 4  |-  ( ( ( Base `  C
)  =  ( Base `  D )  /\  ( Base `  C )  =  ( Base `  D
) )  ->  (
a  e.  ( Base `  C ) ,  b  e.  ( Base `  C
)  |->  { f  e.  ( a (  Hom  `  D ) b )  |  A. c  e.  ( Base `  D
) Fun  `' (
g  e.  ( c (  Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) } )  =  ( a  e.  ( Base `  D
) ,  b  e.  ( Base `  D
)  |->  { f  e.  ( a (  Hom  `  D ) b )  |  A. c  e.  ( Base `  D
) Fun  `' (
g  e.  ( c (  Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) } ) )
3729, 29, 36syl2anc 643 . . 3  |-  ( ph  ->  ( a  e.  (
Base `  C ) ,  b  e.  ( Base `  C )  |->  { f  e.  ( a (  Hom  `  D
) b )  | 
A. c  e.  (
Base `  D ) Fun  `' ( g  e.  ( c (  Hom  `  D ) a ) 
|->  ( f ( <.
c ,  a >.
(comp `  D )
b ) g ) ) } )  =  ( a  e.  (
Base `  D ) ,  b  e.  ( Base `  D )  |->  { f  e.  ( a (  Hom  `  D
) b )  | 
A. c  e.  (
Base `  D ) Fun  `' ( g  e.  ( c (  Hom  `  D ) a ) 
|->  ( f ( <.
c ,  a >.
(comp `  D )
b ) g ) ) } ) )
3835, 37eqtrd 2468 . 2  |-  ( ph  ->  ( a  e.  (
Base `  C ) ,  b  e.  ( Base `  C )  |->  { f  e.  ( a (  Hom  `  C
) b )  | 
A. c  e.  (
Base `  C ) Fun  `' ( g  e.  ( c (  Hom  `  C ) a ) 
|->  ( f ( <.
c ,  a >.
(comp `  C )
b ) g ) ) } )  =  ( a  e.  (
Base `  D ) ,  b  e.  ( Base `  D )  |->  { f  e.  ( a (  Hom  `  D
) b )  | 
A. c  e.  (
Base `  D ) Fun  `' ( g  e.  ( c (  Hom  `  D ) a ) 
|->  ( f ( <.
c ,  a >.
(comp `  D )
b ) g ) ) } ) )
39 eqid 2436 . . 3  |-  (Mono `  C )  =  (Mono `  C )
40 monpropd.c . . 3  |-  ( ph  ->  C  e.  Cat )
411, 2, 10, 39, 40monfval 13958 . 2  |-  ( ph  ->  (Mono `  C )  =  ( a  e.  ( Base `  C
) ,  b  e.  ( Base `  C
)  |->  { f  e.  ( a (  Hom  `  C ) b )  |  A. c  e.  ( Base `  C
) Fun  `' (
g  e.  ( c (  Hom  `  C
) a )  |->  ( f ( <. c ,  a >. (comp `  C ) b ) g ) ) } ) )
42 eqid 2436 . . 3  |-  ( Base `  D )  =  (
Base `  D )
43 eqid 2436 . . 3  |-  (Mono `  D )  =  (Mono `  D )
44 monpropd.d . . 3  |-  ( ph  ->  D  e.  Cat )
4542, 3, 11, 43, 44monfval 13958 . 2  |-  ( ph  ->  (Mono `  D )  =  ( a  e.  ( Base `  D
) ,  b  e.  ( Base `  D
)  |->  { f  e.  ( a (  Hom  `  D ) b )  |  A. c  e.  ( Base `  D
) Fun  `' (
g  e.  ( c (  Hom  `  D
) a )  |->  ( f ( <. c ,  a >. (comp `  D ) b ) g ) ) } ) )
4638, 41, 453eqtr4d 2478 1  |-  ( ph  ->  (Mono `  C )  =  (Mono `  D )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   {crab 2709   <.cop 3817    e. cmpt 4266   `'ccnv 4877   Fun wfun 5448   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   Basecbs 13469    Hom chom 13540  compcco 13541   Catccat 13889    Homf chomf 13891  compfccomf 13892  Monocmon 13954
This theorem is referenced by:  oppcepi  13965
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-homf 13895  df-comf 13896  df-mon 13956
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