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Theorem oppcmon 17720
Description: A monomorphism in the opposite category is an epimorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
oppcmon.o 𝑂 = (oppCatβ€˜πΆ)
oppcmon.c (πœ‘ β†’ 𝐢 ∈ Cat)
oppcmon.m 𝑀 = (Monoβ€˜π‘‚)
oppcmon.e 𝐸 = (Epiβ€˜πΆ)
Assertion
Ref Expression
oppcmon (πœ‘ β†’ (π‘‹π‘€π‘Œ) = (π‘ŒπΈπ‘‹))

Proof of Theorem oppcmon
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 oppcmon.e . . . 4 𝐸 = (Epiβ€˜πΆ)
2 oppcmon.c . . . . 5 (πœ‘ β†’ 𝐢 ∈ Cat)
3 fveq2 6897 . . . . . . . . . 10 (𝑐 = 𝐢 β†’ (oppCatβ€˜π‘) = (oppCatβ€˜πΆ))
4 oppcmon.o . . . . . . . . . 10 𝑂 = (oppCatβ€˜πΆ)
53, 4eqtr4di 2786 . . . . . . . . 9 (𝑐 = 𝐢 β†’ (oppCatβ€˜π‘) = 𝑂)
65fveq2d 6901 . . . . . . . 8 (𝑐 = 𝐢 β†’ (Monoβ€˜(oppCatβ€˜π‘)) = (Monoβ€˜π‘‚))
7 oppcmon.m . . . . . . . 8 𝑀 = (Monoβ€˜π‘‚)
86, 7eqtr4di 2786 . . . . . . 7 (𝑐 = 𝐢 β†’ (Monoβ€˜(oppCatβ€˜π‘)) = 𝑀)
98tposeqd 8234 . . . . . 6 (𝑐 = 𝐢 β†’ tpos (Monoβ€˜(oppCatβ€˜π‘)) = tpos 𝑀)
10 df-epi 17713 . . . . . 6 Epi = (𝑐 ∈ Cat ↦ tpos (Monoβ€˜(oppCatβ€˜π‘)))
117fvexi 6911 . . . . . . 7 𝑀 ∈ V
1211tposex 8265 . . . . . 6 tpos 𝑀 ∈ V
139, 10, 12fvmpt 7005 . . . . 5 (𝐢 ∈ Cat β†’ (Epiβ€˜πΆ) = tpos 𝑀)
142, 13syl 17 . . . 4 (πœ‘ β†’ (Epiβ€˜πΆ) = tpos 𝑀)
151, 14eqtrid 2780 . . 3 (πœ‘ β†’ 𝐸 = tpos 𝑀)
1615oveqd 7437 . 2 (πœ‘ β†’ (π‘ŒπΈπ‘‹) = (π‘Œtpos 𝑀𝑋))
17 ovtpos 8246 . 2 (π‘Œtpos 𝑀𝑋) = (π‘‹π‘€π‘Œ)
1816, 17eqtr2di 2785 1 (πœ‘ β†’ (π‘‹π‘€π‘Œ) = (π‘ŒπΈπ‘‹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1534   ∈ wcel 2099  β€˜cfv 6548  (class class class)co 7420  tpos ctpos 8230  Catccat 17643  oppCatcoppc 17690  Monocmon 17710  Epicepi 17711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-fv 6556  df-ov 7423  df-tpos 8231  df-epi 17713
This theorem is referenced by:  oppcepi  17721  isepi  17722  epii  17725  sectepi  17766  episect  17767  fthepi  17916
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