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Theorem oppcmon 13893
Description: A monomorphism in the opposite category is an epimorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
oppcmon.o  |-  O  =  (oppCat `  C )
oppcmon.c  |-  ( ph  ->  C  e.  Cat )
oppcmon.m  |-  M  =  (Mono `  O )
oppcmon.e  |-  E  =  (Epi `  C )
Assertion
Ref Expression
oppcmon  |-  ( ph  ->  ( X M Y )  =  ( Y E X ) )

Proof of Theorem oppcmon
Dummy variable  c is distinct from all other variables.
StepHypRef Expression
1 oppcmon.e . . . 4  |-  E  =  (Epi `  C )
2 oppcmon.c . . . . 5  |-  ( ph  ->  C  e.  Cat )
3 fveq2 5670 . . . . . . . . . 10  |-  ( c  =  C  ->  (oppCat `  c )  =  (oppCat `  C ) )
4 oppcmon.o . . . . . . . . . 10  |-  O  =  (oppCat `  C )
53, 4syl6eqr 2439 . . . . . . . . 9  |-  ( c  =  C  ->  (oppCat `  c )  =  O )
65fveq2d 5674 . . . . . . . 8  |-  ( c  =  C  ->  (Mono `  (oppCat `  c )
)  =  (Mono `  O ) )
7 oppcmon.m . . . . . . . 8  |-  M  =  (Mono `  O )
86, 7syl6eqr 2439 . . . . . . 7  |-  ( c  =  C  ->  (Mono `  (oppCat `  c )
)  =  M )
98tposeqd 6420 . . . . . 6  |-  ( c  =  C  -> tpos  (Mono `  (oppCat `  c ) )  = tpos  M )
10 df-epi 13886 . . . . . 6  |- Epi  =  ( c  e.  Cat  |-> tpos  (Mono `  (oppCat `  c )
) )
11 fvex 5684 . . . . . . . 8  |-  (Mono `  O )  e.  _V
127, 11eqeltri 2459 . . . . . . 7  |-  M  e. 
_V
1312tposex 6451 . . . . . 6  |- tpos  M  e. 
_V
149, 10, 13fvmpt 5747 . . . . 5  |-  ( C  e.  Cat  ->  (Epi `  C )  = tpos  M
)
152, 14syl 16 . . . 4  |-  ( ph  ->  (Epi `  C )  = tpos  M )
161, 15syl5eq 2433 . . 3  |-  ( ph  ->  E  = tpos  M )
1716oveqd 6039 . 2  |-  ( ph  ->  ( Y E X )  =  ( Ytpos 
M X ) )
18 ovtpos 6432 . 2  |-  ( Ytpos 
M X )  =  ( X M Y )
1917, 18syl6req 2438 1  |-  ( ph  ->  ( X M Y )  =  ( Y E X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   _Vcvv 2901   ` cfv 5396  (class class class)co 6022  tpos ctpos 6416   Catccat 13818  oppCatcoppc 13866  Monocmon 13883  Epicepi 13884
This theorem is referenced by:  oppcepi  13894  isepi  13895  epii  13898  sectepi  13934  episect  13935  fthepi  14054
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-fv 5404  df-ov 6025  df-tpos 6417  df-epi 13886
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