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Theorem oppcmon 17786
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 6907 . . . . . . . . . 10 (𝑐 = 𝐶 → (oppCat‘𝑐) = (oppCat‘𝐶))
4 oppcmon.o . . . . . . . . . 10 𝑂 = (oppCat‘𝐶)
53, 4eqtr4di 2793 . . . . . . . . 9 (𝑐 = 𝐶 → (oppCat‘𝑐) = 𝑂)
65fveq2d 6911 . . . . . . . 8 (𝑐 = 𝐶 → (Mono‘(oppCat‘𝑐)) = (Mono‘𝑂))
7 oppcmon.m . . . . . . . 8 𝑀 = (Mono‘𝑂)
86, 7eqtr4di 2793 . . . . . . 7 (𝑐 = 𝐶 → (Mono‘(oppCat‘𝑐)) = 𝑀)
98tposeqd 8253 . . . . . 6 (𝑐 = 𝐶 → tpos (Mono‘(oppCat‘𝑐)) = tpos 𝑀)
10 df-epi 17779 . . . . . 6 Epi = (𝑐 ∈ Cat ↦ tpos (Mono‘(oppCat‘𝑐)))
117fvexi 6921 . . . . . . 7 𝑀 ∈ V
1211tposex 8284 . . . . . 6 tpos 𝑀 ∈ V
139, 10, 12fvmpt 7016 . . . . 5 (𝐶 ∈ Cat → (Epi‘𝐶) = tpos 𝑀)
142, 13syl 17 . . . 4 (𝜑 → (Epi‘𝐶) = tpos 𝑀)
151, 14eqtrid 2787 . . 3 (𝜑𝐸 = tpos 𝑀)
1615oveqd 7448 . 2 (𝜑 → (𝑌𝐸𝑋) = (𝑌tpos 𝑀𝑋))
17 ovtpos 8265 . 2 (𝑌tpos 𝑀𝑋) = (𝑋𝑀𝑌)
1816, 17eqtr2di 2792 1 (𝜑 → (𝑋𝑀𝑌) = (𝑌𝐸𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cfv 6563  (class class class)co 7431  tpos ctpos 8249  Catccat 17709  oppCatcoppc 17756  Monocmon 17776  Epicepi 17777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-ov 7434  df-tpos 8250  df-epi 17779
This theorem is referenced by:  oppcepi  17787  isepi  17788  epii  17791  sectepi  17832  episect  17833  fthepi  17982
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