Theorem oppcmon 17450

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 6774	. . . . . . . . . 10 ⊢ (𝑐 = 𝐶 → (oppCat‘𝑐) = (oppCat‘𝐶))
4		oppcmon.o	. . . . . . . . . 10 ⊢ 𝑂 = (oppCat‘𝐶)
5	3, 4	eqtr4di 2796	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (oppCat‘𝑐) = 𝑂)
6	5	fveq2d 6778	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Mono‘(oppCat‘𝑐)) = (Mono‘𝑂))
7		oppcmon.m	. . . . . . . 8 ⊢ 𝑀 = (Mono‘𝑂)
8	6, 7	eqtr4di 2796	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Mono‘(oppCat‘𝑐)) = 𝑀)
9	8	tposeqd 8045	. . . . . 6 ⊢ (𝑐 = 𝐶 → tpos (Mono‘(oppCat‘𝑐)) = tpos 𝑀)
10		df-epi 17443	. . . . . 6 ⊢ Epi = (𝑐 ∈ Cat ↦ tpos (Mono‘(oppCat‘𝑐)))
11	7	fvexi 6788	. . . . . . 7 ⊢ 𝑀 ∈ V
12	11	tposex 8076	. . . . . 6 ⊢ tpos 𝑀 ∈ V
13	9, 10, 12	fvmpt 6875	. . . . 5 ⊢ (𝐶 ∈ Cat → (Epi‘𝐶) = tpos 𝑀)
14	2, 13	syl 17	. . . 4 ⊢ (𝜑 → (Epi‘𝐶) = tpos 𝑀)
15	1, 14	eqtrid 2790	. . 3 ⊢ (𝜑 → 𝐸 = tpos 𝑀)
16	15	oveqd 7292	. 2 ⊢ (𝜑 → (𝑌𝐸𝑋) = (𝑌tpos 𝑀𝑋))
17		ovtpos 8057	. 2 ⊢ (𝑌tpos 𝑀𝑋) = (𝑋𝑀𝑌)
18	16, 17	eqtr2di 2795	1 ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑌𝐸𝑋))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  ‘cfv 6433  (class class class)co 7275  tpos ctpos 8041  Catccat 17373  oppCatcoppc 17420  Monocmon 17440  Epicepi 17441

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441  df-ov 7278  df-tpos 8042  df-epi 17443

This theorem is referenced by:  oppcepi  17451  isepi  17452  epii  17455  sectepi  17496  episect  17497  fthepi  17644