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Theorem oppcmon 17629
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 6846 . . . . . . . . . 10 (𝑐 = 𝐢 β†’ (oppCatβ€˜π‘) = (oppCatβ€˜πΆ))
4 oppcmon.o . . . . . . . . . 10 𝑂 = (oppCatβ€˜πΆ)
53, 4eqtr4di 2791 . . . . . . . . 9 (𝑐 = 𝐢 β†’ (oppCatβ€˜π‘) = 𝑂)
65fveq2d 6850 . . . . . . . 8 (𝑐 = 𝐢 β†’ (Monoβ€˜(oppCatβ€˜π‘)) = (Monoβ€˜π‘‚))
7 oppcmon.m . . . . . . . 8 𝑀 = (Monoβ€˜π‘‚)
86, 7eqtr4di 2791 . . . . . . 7 (𝑐 = 𝐢 β†’ (Monoβ€˜(oppCatβ€˜π‘)) = 𝑀)
98tposeqd 8164 . . . . . 6 (𝑐 = 𝐢 β†’ tpos (Monoβ€˜(oppCatβ€˜π‘)) = tpos 𝑀)
10 df-epi 17622 . . . . . 6 Epi = (𝑐 ∈ Cat ↦ tpos (Monoβ€˜(oppCatβ€˜π‘)))
117fvexi 6860 . . . . . . 7 𝑀 ∈ V
1211tposex 8195 . . . . . 6 tpos 𝑀 ∈ V
139, 10, 12fvmpt 6952 . . . . 5 (𝐢 ∈ Cat β†’ (Epiβ€˜πΆ) = tpos 𝑀)
142, 13syl 17 . . . 4 (πœ‘ β†’ (Epiβ€˜πΆ) = tpos 𝑀)
151, 14eqtrid 2785 . . 3 (πœ‘ β†’ 𝐸 = tpos 𝑀)
1615oveqd 7378 . 2 (πœ‘ β†’ (π‘ŒπΈπ‘‹) = (π‘Œtpos 𝑀𝑋))
17 ovtpos 8176 . 2 (π‘Œtpos 𝑀𝑋) = (π‘‹π‘€π‘Œ)
1816, 17eqtr2di 2790 1 (πœ‘ β†’ (π‘‹π‘€π‘Œ) = (π‘ŒπΈπ‘‹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  β€˜cfv 6500  (class class class)co 7361  tpos ctpos 8160  Catccat 17552  oppCatcoppc 17599  Monocmon 17619  Epicepi 17620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-fv 6508  df-ov 7364  df-tpos 8161  df-epi 17622
This theorem is referenced by:  oppcepi  17630  isepi  17631  epii  17634  sectepi  17675  episect  17676  fthepi  17823
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