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Theorem oppcmon 14676
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 5690 . . . . . . . . . 10  |-  ( c  =  C  ->  (oppCat `  c )  =  (oppCat `  C ) )
4 oppcmon.o . . . . . . . . . 10  |-  O  =  (oppCat `  C )
53, 4syl6eqr 2492 . . . . . . . . 9  |-  ( c  =  C  ->  (oppCat `  c )  =  O )
65fveq2d 5694 . . . . . . . 8  |-  ( c  =  C  ->  (Mono `  (oppCat `  c )
)  =  (Mono `  O ) )
7 oppcmon.m . . . . . . . 8  |-  M  =  (Mono `  O )
86, 7syl6eqr 2492 . . . . . . 7  |-  ( c  =  C  ->  (Mono `  (oppCat `  c )
)  =  M )
98tposeqd 6747 . . . . . 6  |-  ( c  =  C  -> tpos  (Mono `  (oppCat `  c ) )  = tpos  M )
10 df-epi 14669 . . . . . 6  |- Epi  =  ( c  e.  Cat  |-> tpos  (Mono `  (oppCat `  c )
) )
11 fvex 5700 . . . . . . . 8  |-  (Mono `  O )  e.  _V
127, 11eqeltri 2512 . . . . . . 7  |-  M  e. 
_V
1312tposex 6778 . . . . . 6  |- tpos  M  e. 
_V
149, 10, 13fvmpt 5773 . . . . 5  |-  ( C  e.  Cat  ->  (Epi `  C )  = tpos  M
)
152, 14syl 16 . . . 4  |-  ( ph  ->  (Epi `  C )  = tpos  M )
161, 15syl5eq 2486 . . 3  |-  ( ph  ->  E  = tpos  M )
1716oveqd 6107 . 2  |-  ( ph  ->  ( Y E X )  =  ( Ytpos 
M X ) )
18 ovtpos 6759 . 2  |-  ( Ytpos 
M X )  =  ( X M Y )
1917, 18syl6req 2491 1  |-  ( ph  ->  ( X M Y )  =  ( Y E X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   _Vcvv 2971   ` cfv 5417  (class class class)co 6090  tpos ctpos 6743   Catccat 14601  oppCatcoppc 14649  Monocmon 14666  Epicepi 14667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-fv 5425  df-ov 6093  df-tpos 6744  df-epi 14669
This theorem is referenced by:  oppcepi  14677  isepi  14678  epii  14681  sectepi  14717  episect  14718  fthepi  14837
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