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Theorem oppcmon 17782
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 6906 . . . . . . . . . 10 (𝑐 = 𝐶 → (oppCat‘𝑐) = (oppCat‘𝐶))
4 oppcmon.o . . . . . . . . . 10 𝑂 = (oppCat‘𝐶)
53, 4eqtr4di 2795 . . . . . . . . 9 (𝑐 = 𝐶 → (oppCat‘𝑐) = 𝑂)
65fveq2d 6910 . . . . . . . 8 (𝑐 = 𝐶 → (Mono‘(oppCat‘𝑐)) = (Mono‘𝑂))
7 oppcmon.m . . . . . . . 8 𝑀 = (Mono‘𝑂)
86, 7eqtr4di 2795 . . . . . . 7 (𝑐 = 𝐶 → (Mono‘(oppCat‘𝑐)) = 𝑀)
98tposeqd 8254 . . . . . 6 (𝑐 = 𝐶 → tpos (Mono‘(oppCat‘𝑐)) = tpos 𝑀)
10 df-epi 17775 . . . . . 6 Epi = (𝑐 ∈ Cat ↦ tpos (Mono‘(oppCat‘𝑐)))
117fvexi 6920 . . . . . . 7 𝑀 ∈ V
1211tposex 8285 . . . . . 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 2789 . . 3 (𝜑𝐸 = tpos 𝑀)
1615oveqd 7448 . 2 (𝜑 → (𝑌𝐸𝑋) = (𝑌tpos 𝑀𝑋))
17 ovtpos 8266 . 2 (𝑌tpos 𝑀𝑋) = (𝑋𝑀𝑌)
1816, 17eqtr2di 2794 1 (𝜑 → (𝑋𝑀𝑌) = (𝑌𝐸𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cfv 6561  (class class class)co 7431  tpos ctpos 8250  Catccat 17707  oppCatcoppc 17754  Monocmon 17772  Epicepi 17773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-ov 7434  df-tpos 8251  df-epi 17775
This theorem is referenced by:  oppcepi  17783  isepi  17784  epii  17787  sectepi  17828  episect  17829  fthepi  17975
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