Theorem oppcmon 16221

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 6103	. . . . . . . . . 10 ⊢ (𝑐 = 𝐶 → (oppCat‘𝑐) = (oppCat‘𝐶))
4		oppcmon.o	. . . . . . . . . 10 ⊢ 𝑂 = (oppCat‘𝐶)
5	3, 4	syl6eqr 2662	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (oppCat‘𝑐) = 𝑂)
6	5	fveq2d 6107	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Mono‘(oppCat‘𝑐)) = (Mono‘𝑂))
7		oppcmon.m	. . . . . . . 8 ⊢ 𝑀 = (Mono‘𝑂)
8	6, 7	syl6eqr 2662	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Mono‘(oppCat‘𝑐)) = 𝑀)
9	8	tposeqd 7242	. . . . . 6 ⊢ (𝑐 = 𝐶 → tpos (Mono‘(oppCat‘𝑐)) = tpos 𝑀)
10		df-epi 16214	. . . . . 6 ⊢ Epi = (𝑐 ∈ Cat ↦ tpos (Mono‘(oppCat‘𝑐)))
11		fvex 6113	. . . . . . . 8 ⊢ (Mono‘𝑂) ∈ V
12	7, 11	eqeltri 2684	. . . . . . 7 ⊢ 𝑀 ∈ V
13	12	tposex 7273	. . . . . 6 ⊢ tpos 𝑀 ∈ V
14	9, 10, 13	fvmpt 6191	. . . . 5 ⊢ (𝐶 ∈ Cat → (Epi‘𝐶) = tpos 𝑀)
15	2, 14	syl 17	. . . 4 ⊢ (𝜑 → (Epi‘𝐶) = tpos 𝑀)
16	1, 15	syl5eq 2656	. . 3 ⊢ (𝜑 → 𝐸 = tpos 𝑀)
17	16	oveqd 6566	. 2 ⊢ (𝜑 → (𝑌𝐸𝑋) = (𝑌tpos 𝑀𝑋))
18		ovtpos 7254	. 2 ⊢ (𝑌tpos 𝑀𝑋) = (𝑋𝑀𝑌)
19	17, 18	syl6req 2661	1 ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑌𝐸𝑋))

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ‘cfv 5804  (class class class)co 6549  tpos ctpos 7238  Catccat 16148  oppCatcoppc 16194  Monocmon 16211  Epicepi 16212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-ov 6552  df-tpos 7239  df-epi 16214

This theorem is referenced by:  oppcepi  16222  isepi  16223  epii  16226  sectepi  16267  episect  16268  fthepi  16411