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Theorem oppcmon 13641
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 5525 . . . . . . . . . 10  |-  ( c  =  C  ->  (oppCat `  c )  =  (oppCat `  C ) )
4 oppcmon.o . . . . . . . . . 10  |-  O  =  (oppCat `  C )
53, 4syl6eqr 2333 . . . . . . . . 9  |-  ( c  =  C  ->  (oppCat `  c )  =  O )
65fveq2d 5529 . . . . . . . 8  |-  ( c  =  C  ->  (Mono `  (oppCat `  c )
)  =  (Mono `  O ) )
7 oppcmon.m . . . . . . . 8  |-  M  =  (Mono `  O )
86, 7syl6eqr 2333 . . . . . . 7  |-  ( c  =  C  ->  (Mono `  (oppCat `  c )
)  =  M )
9 tposeq 6236 . . . . . . 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 13634 . . . . . 6  |- Epi  =  ( c  e.  Cat  |-> tpos  (Mono `  (oppCat `  c )
) )
12 fvex 5539 . . . . . . . 8  |-  (Mono `  O )  e.  _V
137, 12eqeltri 2353 . . . . . . 7  |-  M  e. 
_V
1413tposex 6268 . . . . . 6  |- tpos  M  e. 
_V
1510, 11, 14fvmpt 5602 . . . . 5  |-  ( C  e.  Cat  ->  (Epi `  C )  = tpos  M
)
162, 15syl 15 . . . 4  |-  ( ph  ->  (Epi `  C )  = tpos  M )
171, 16syl5eq 2327 . . 3  |-  ( ph  ->  E  = tpos  M )
1817oveqd 5875 . 2  |-  ( ph  ->  ( Y E X )  =  ( Ytpos 
M X ) )
19 ovtpos 6249 . 2  |-  ( Ytpos 
M X )  =  ( X M Y )
2018, 19syl6req 2332 1  |-  ( ph  ->  ( X M Y )  =  ( Y E X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788   ` cfv 5255  (class class class)co 5858  tpos ctpos 6233   Catccat 13566  oppCatcoppc 13614  Monocmon 13631  Epicepi 13632
This theorem is referenced by:  oppcepi  13642  isepi  13643  epii  13646  sectepi  13682  episect  13683  fthepi  13802
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-ov 5861  df-tpos 6234  df-epi 13634
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