Theorem oppcmon 17833

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	⊢ (𝜑 → 𝐶 ∈ 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.

Step	Hyp	Ref	Expression
1		oppcmon.e	. . . 4 ⊢ 𝐸 = (Epi‘𝐶)
2		oppcmon.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
3		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑐 = 𝐶 → (oppCat‘𝑐) = (oppCat‘𝐶))
4		oppcmon.o	. . . . . . . . . 10 ⊢ 𝑂 = (oppCat‘𝐶)
5	3, 4	eqtr4di 2815	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (oppCat‘𝑐) = 𝑂)
6	5	fveq2d 6886	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Mono‘(oppCat‘𝑐)) = (Mono‘𝑂))
7		oppcmon.m	. . . . . . . 8 ⊢ 𝑀 = (Mono‘𝑂)
8	6, 7	eqtr4di 2815	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Mono‘(oppCat‘𝑐)) = 𝑀)
9	8	tposeqd 8231	. . . . . 6 ⊢ (𝑐 = 𝐶 → tpos (Mono‘(oppCat‘𝑐)) = tpos 𝑀)
10		df-epi 17826	. . . . . 6 ⊢ Epi = (𝑐 ∈ Cat ↦ tpos (Mono‘(oppCat‘𝑐)))
11	7	fvexi 6896	. . . . . . 7 ⊢ 𝑀 ∈ V
12	11	tposex 8262	. . . . . 6 ⊢ tpos 𝑀 ∈ V
13	9, 10, 12	fvmpt 6990	. . . . 5 ⊢ (𝐶 ∈ Cat → (Epi‘𝐶) = tpos 𝑀)
14	2, 13	syl 18	. . . 4 ⊢ (𝜑 → (Epi‘𝐶) = tpos 𝑀)
15	1, 14	eqtrid 2809	. . 3 ⊢ (𝜑 → 𝐸 = tpos 𝑀)
16	15	oveqd 7434	. 2 ⊢ (𝜑 → (𝑌𝐸𝑋) = (𝑌tpos 𝑀𝑋))
17		ovtpos 8243	. 2 ⊢ (𝑌tpos 𝑀𝑋) = (𝑋𝑀𝑌)
18	16, 17	eqtr2di 2814	1 ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑌𝐸𝑋))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ‘cfv 6537  (class class class)co 7417  tpos ctpos 8227  Catccat 17758  oppCatcoppc 17805  Monocmon 17823  Epicepi 17824

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-fv 6545  df-ov 7420  df-tpos 8228  df-epi 17826

This theorem is used by:  oppcepi  17834  isepi  17835  epii  17838  sectepi  17879  episect  17880  fthepi  18025