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Theorem oppcmon 16599
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 6352 . . . . . . . . . 10 (𝑐 = 𝐶 → (oppCat‘𝑐) = (oppCat‘𝐶))
4 oppcmon.o . . . . . . . . . 10 𝑂 = (oppCat‘𝐶)
53, 4syl6eqr 2812 . . . . . . . . 9 (𝑐 = 𝐶 → (oppCat‘𝑐) = 𝑂)
65fveq2d 6356 . . . . . . . 8 (𝑐 = 𝐶 → (Mono‘(oppCat‘𝑐)) = (Mono‘𝑂))
7 oppcmon.m . . . . . . . 8 𝑀 = (Mono‘𝑂)
86, 7syl6eqr 2812 . . . . . . 7 (𝑐 = 𝐶 → (Mono‘(oppCat‘𝑐)) = 𝑀)
98tposeqd 7524 . . . . . 6 (𝑐 = 𝐶 → tpos (Mono‘(oppCat‘𝑐)) = tpos 𝑀)
10 df-epi 16592 . . . . . 6 Epi = (𝑐 ∈ Cat ↦ tpos (Mono‘(oppCat‘𝑐)))
11 fvex 6362 . . . . . . . 8 (Mono‘𝑂) ∈ V
127, 11eqeltri 2835 . . . . . . 7 𝑀 ∈ V
1312tposex 7555 . . . . . 6 tpos 𝑀 ∈ V
149, 10, 13fvmpt 6444 . . . . 5 (𝐶 ∈ Cat → (Epi‘𝐶) = tpos 𝑀)
152, 14syl 17 . . . 4 (𝜑 → (Epi‘𝐶) = tpos 𝑀)
161, 15syl5eq 2806 . . 3 (𝜑𝐸 = tpos 𝑀)
1716oveqd 6830 . 2 (𝜑 → (𝑌𝐸𝑋) = (𝑌tpos 𝑀𝑋))
18 ovtpos 7536 . 2 (𝑌tpos 𝑀𝑋) = (𝑋𝑀𝑌)
1917, 18syl6req 2811 1 (𝜑 → (𝑋𝑀𝑌) = (𝑌𝐸𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  Vcvv 3340  cfv 6049  (class class class)co 6813  tpos ctpos 7520  Catccat 16526  oppCatcoppc 16572  Monocmon 16589  Epicepi 16590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057  df-ov 6816  df-tpos 7521  df-epi 16592
This theorem is referenced by:  oppcepi  16600  isepi  16601  epii  16604  sectepi  16645  episect  16646  fthepi  16789
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