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Theorem oppcmon 16445
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 (𝜑𝐶 ∈ Cat)
oppcmon.m 𝑀 = (Mono‘𝑂)
oppcmon.e 𝐸 = (Epi‘𝐶)
Assertion
Ref Expression
oppcmon (𝜑 → (𝑋𝑀𝑌) = (𝑌𝐸𝑋))

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 (𝜑𝐶 ∈ Cat)
3 fveq2 6229 . . . . . . . . . 10 (𝑐 = 𝐶 → (oppCat‘𝑐) = (oppCat‘𝐶))
4 oppcmon.o . . . . . . . . . 10 𝑂 = (oppCat‘𝐶)
53, 4syl6eqr 2703 . . . . . . . . 9 (𝑐 = 𝐶 → (oppCat‘𝑐) = 𝑂)
65fveq2d 6233 . . . . . . . 8 (𝑐 = 𝐶 → (Mono‘(oppCat‘𝑐)) = (Mono‘𝑂))
7 oppcmon.m . . . . . . . 8 𝑀 = (Mono‘𝑂)
86, 7syl6eqr 2703 . . . . . . 7 (𝑐 = 𝐶 → (Mono‘(oppCat‘𝑐)) = 𝑀)
98tposeqd 7400 . . . . . 6 (𝑐 = 𝐶 → tpos (Mono‘(oppCat‘𝑐)) = tpos 𝑀)
10 df-epi 16438 . . . . . 6 Epi = (𝑐 ∈ Cat ↦ tpos (Mono‘(oppCat‘𝑐)))
11 fvex 6239 . . . . . . . 8 (Mono‘𝑂) ∈ V
127, 11eqeltri 2726 . . . . . . 7 𝑀 ∈ V
1312tposex 7431 . . . . . 6 tpos 𝑀 ∈ V
149, 10, 13fvmpt 6321 . . . . 5 (𝐶 ∈ Cat → (Epi‘𝐶) = tpos 𝑀)
152, 14syl 17 . . . 4 (𝜑 → (Epi‘𝐶) = tpos 𝑀)
161, 15syl5eq 2697 . . 3 (𝜑𝐸 = tpos 𝑀)
1716oveqd 6707 . 2 (𝜑 → (𝑌𝐸𝑋) = (𝑌tpos 𝑀𝑋))
18 ovtpos 7412 . 2 (𝑌tpos 𝑀𝑋) = (𝑋𝑀𝑌)
1917, 18syl6req 2702 1 (𝜑 → (𝑋𝑀𝑌) = (𝑌𝐸𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  Vcvv 3231  cfv 5926  (class class class)co 6690  tpos ctpos 7396  Catccat 16372  oppCatcoppc 16418  Monocmon 16435  Epicepi 16436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-ov 6693  df-tpos 7397  df-epi 16438
This theorem is referenced by:  oppcepi  16446  isepi  16447  epii  16450  sectepi  16491  episect  16492  fthepi  16635
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