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Theorem oppcmon 13657
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 5541 . . . . . . . . . 10  |-  ( c  =  C  ->  (oppCat `  c )  =  (oppCat `  C ) )
4 oppcmon.o . . . . . . . . . 10  |-  O  =  (oppCat `  C )
53, 4syl6eqr 2346 . . . . . . . . 9  |-  ( c  =  C  ->  (oppCat `  c )  =  O )
65fveq2d 5545 . . . . . . . 8  |-  ( c  =  C  ->  (Mono `  (oppCat `  c )
)  =  (Mono `  O ) )
7 oppcmon.m . . . . . . . 8  |-  M  =  (Mono `  O )
86, 7syl6eqr 2346 . . . . . . 7  |-  ( c  =  C  ->  (Mono `  (oppCat `  c )
)  =  M )
9 tposeq 6252 . . . . . . 7  |-  ( (Mono `  (oppCat `  c )
)  =  M  -> tpos  (Mono `  (oppCat `  c
) )  = tpos  M
)
108, 9syl 15 . . . . . 6  |-  ( c  =  C  -> tpos  (Mono `  (oppCat `  c ) )  = tpos  M )
11 df-epi 13650 . . . . . 6  |- Epi  =  ( c  e.  Cat  |-> tpos  (Mono `  (oppCat `  c )
) )
12 fvex 5555 . . . . . . . 8  |-  (Mono `  O )  e.  _V
137, 12eqeltri 2366 . . . . . . 7  |-  M  e. 
_V
1413tposex 6284 . . . . . 6  |- tpos  M  e. 
_V
1510, 11, 14fvmpt 5618 . . . . 5  |-  ( C  e.  Cat  ->  (Epi `  C )  = tpos  M
)
162, 15syl 15 . . . 4  |-  ( ph  ->  (Epi `  C )  = tpos  M )
171, 16syl5eq 2340 . . 3  |-  ( ph  ->  E  = tpos  M )
1817oveqd 5891 . 2  |-  ( ph  ->  ( Y E X )  =  ( Ytpos 
M X ) )
19 ovtpos 6265 . 2  |-  ( Ytpos 
M X )  =  ( X M Y )
2018, 19syl6req 2345 1  |-  ( ph  ->  ( X M Y )  =  ( Y E X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801   ` cfv 5271  (class class class)co 5874  tpos ctpos 6249   Catccat 13582  oppCatcoppc 13630  Monocmon 13647  Epicepi 13648
This theorem is referenced by:  oppcepi  13658  isepi  13659  epii  13662  sectepi  13698  episect  13699  fthepi  13818
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-ov 5877  df-tpos 6250  df-epi 13650
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