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Theorem oppcmon 16997
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 6651 . . . . . . . . . 10 (𝑐 = 𝐶 → (oppCat‘𝑐) = (oppCat‘𝐶))
4 oppcmon.o . . . . . . . . . 10 𝑂 = (oppCat‘𝐶)
53, 4syl6eqr 2877 . . . . . . . . 9 (𝑐 = 𝐶 → (oppCat‘𝑐) = 𝑂)
65fveq2d 6655 . . . . . . . 8 (𝑐 = 𝐶 → (Mono‘(oppCat‘𝑐)) = (Mono‘𝑂))
7 oppcmon.m . . . . . . . 8 𝑀 = (Mono‘𝑂)
86, 7syl6eqr 2877 . . . . . . 7 (𝑐 = 𝐶 → (Mono‘(oppCat‘𝑐)) = 𝑀)
98tposeqd 7878 . . . . . 6 (𝑐 = 𝐶 → tpos (Mono‘(oppCat‘𝑐)) = tpos 𝑀)
10 df-epi 16990 . . . . . 6 Epi = (𝑐 ∈ Cat ↦ tpos (Mono‘(oppCat‘𝑐)))
117fvexi 6665 . . . . . . 7 𝑀 ∈ V
1211tposex 7909 . . . . . 6 tpos 𝑀 ∈ V
139, 10, 12fvmpt 6749 . . . . 5 (𝐶 ∈ Cat → (Epi‘𝐶) = tpos 𝑀)
142, 13syl 17 . . . 4 (𝜑 → (Epi‘𝐶) = tpos 𝑀)
151, 14syl5eq 2871 . . 3 (𝜑𝐸 = tpos 𝑀)
1615oveqd 7155 . 2 (𝜑 → (𝑌𝐸𝑋) = (𝑌tpos 𝑀𝑋))
17 ovtpos 7890 . 2 (𝑌tpos 𝑀𝑋) = (𝑋𝑀𝑌)
1816, 17syl6req 2876 1 (𝜑 → (𝑋𝑀𝑌) = (𝑌𝐸𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  cfv 6336  (class class class)co 7138  tpos ctpos 7874  Catccat 16924  oppCatcoppc 16970  Monocmon 16987  Epicepi 16988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fn 6339  df-fv 6344  df-ov 7141  df-tpos 7875  df-epi 16990
This theorem is referenced by:  oppcepi  16998  isepi  16999  epii  17002  sectepi  17043  episect  17044  fthepi  17187
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