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Theorem oppcmon 13956
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 5720 . . . . . . . . . 10  |-  ( c  =  C  ->  (oppCat `  c )  =  (oppCat `  C ) )
4 oppcmon.o . . . . . . . . . 10  |-  O  =  (oppCat `  C )
53, 4syl6eqr 2485 . . . . . . . . 9  |-  ( c  =  C  ->  (oppCat `  c )  =  O )
65fveq2d 5724 . . . . . . . 8  |-  ( c  =  C  ->  (Mono `  (oppCat `  c )
)  =  (Mono `  O ) )
7 oppcmon.m . . . . . . . 8  |-  M  =  (Mono `  O )
86, 7syl6eqr 2485 . . . . . . 7  |-  ( c  =  C  ->  (Mono `  (oppCat `  c )
)  =  M )
98tposeqd 6474 . . . . . 6  |-  ( c  =  C  -> tpos  (Mono `  (oppCat `  c ) )  = tpos  M )
10 df-epi 13949 . . . . . 6  |- Epi  =  ( c  e.  Cat  |-> tpos  (Mono `  (oppCat `  c )
) )
11 fvex 5734 . . . . . . . 8  |-  (Mono `  O )  e.  _V
127, 11eqeltri 2505 . . . . . . 7  |-  M  e. 
_V
1312tposex 6505 . . . . . 6  |- tpos  M  e. 
_V
149, 10, 13fvmpt 5798 . . . . 5  |-  ( C  e.  Cat  ->  (Epi `  C )  = tpos  M
)
152, 14syl 16 . . . 4  |-  ( ph  ->  (Epi `  C )  = tpos  M )
161, 15syl5eq 2479 . . 3  |-  ( ph  ->  E  = tpos  M )
1716oveqd 6090 . 2  |-  ( ph  ->  ( Y E X )  =  ( Ytpos 
M X ) )
18 ovtpos 6486 . 2  |-  ( Ytpos 
M X )  =  ( X M Y )
1917, 18syl6req 2484 1  |-  ( ph  ->  ( X M Y )  =  ( Y E X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2948   ` cfv 5446  (class class class)co 6073  tpos ctpos 6470   Catccat 13881  oppCatcoppc 13929  Monocmon 13946  Epicepi 13947
This theorem is referenced by:  oppcepi  13957  isepi  13958  epii  13961  sectepi  13997  episect  13998  fthepi  14117
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-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-rab 2706  df-v 2950  df-sbc 3154  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-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-fv 5454  df-ov 6076  df-tpos 6471  df-epi 13949
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