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Theorem oppcmon 17690
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 6882 . . . . . . . . . 10 (𝑐 = 𝐢 β†’ (oppCatβ€˜π‘) = (oppCatβ€˜πΆ))
4 oppcmon.o . . . . . . . . . 10 𝑂 = (oppCatβ€˜πΆ)
53, 4eqtr4di 2782 . . . . . . . . 9 (𝑐 = 𝐢 β†’ (oppCatβ€˜π‘) = 𝑂)
65fveq2d 6886 . . . . . . . 8 (𝑐 = 𝐢 β†’ (Monoβ€˜(oppCatβ€˜π‘)) = (Monoβ€˜π‘‚))
7 oppcmon.m . . . . . . . 8 𝑀 = (Monoβ€˜π‘‚)
86, 7eqtr4di 2782 . . . . . . 7 (𝑐 = 𝐢 β†’ (Monoβ€˜(oppCatβ€˜π‘)) = 𝑀)
98tposeqd 8210 . . . . . 6 (𝑐 = 𝐢 β†’ tpos (Monoβ€˜(oppCatβ€˜π‘)) = tpos 𝑀)
10 df-epi 17683 . . . . . 6 Epi = (𝑐 ∈ Cat ↦ tpos (Monoβ€˜(oppCatβ€˜π‘)))
117fvexi 6896 . . . . . . 7 𝑀 ∈ V
1211tposex 8241 . . . . . 6 tpos 𝑀 ∈ V
139, 10, 12fvmpt 6989 . . . . 5 (𝐢 ∈ Cat β†’ (Epiβ€˜πΆ) = tpos 𝑀)
142, 13syl 17 . . . 4 (πœ‘ β†’ (Epiβ€˜πΆ) = tpos 𝑀)
151, 14eqtrid 2776 . . 3 (πœ‘ β†’ 𝐸 = tpos 𝑀)
1615oveqd 7419 . 2 (πœ‘ β†’ (π‘ŒπΈπ‘‹) = (π‘Œtpos 𝑀𝑋))
17 ovtpos 8222 . 2 (π‘Œtpos 𝑀𝑋) = (π‘‹π‘€π‘Œ)
1816, 17eqtr2di 2781 1 (πœ‘ β†’ (π‘‹π‘€π‘Œ) = (π‘ŒπΈπ‘‹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  β€˜cfv 6534  (class class class)co 7402  tpos ctpos 8206  Catccat 17613  oppCatcoppc 17660  Monocmon 17680  Epicepi 17681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-fv 6542  df-ov 7405  df-tpos 8207  df-epi 17683
This theorem is referenced by:  oppcepi  17691  isepi  17692  epii  17695  sectepi  17736  episect  17737  fthepi  17886
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