Theorem oppcmon 17754

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 6901	. . . . . . . . . 10 ⊢ (𝑐 = 𝐶 → (oppCat‘𝑐) = (oppCat‘𝐶))
4		oppcmon.o	. . . . . . . . . 10 ⊢ 𝑂 = (oppCat‘𝐶)
5	3, 4	eqtr4di 2784	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (oppCat‘𝑐) = 𝑂)
6	5	fveq2d 6905	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Mono‘(oppCat‘𝑐)) = (Mono‘𝑂))
7		oppcmon.m	. . . . . . . 8 ⊢ 𝑀 = (Mono‘𝑂)
8	6, 7	eqtr4di 2784	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Mono‘(oppCat‘𝑐)) = 𝑀)
9	8	tposeqd 8244	. . . . . 6 ⊢ (𝑐 = 𝐶 → tpos (Mono‘(oppCat‘𝑐)) = tpos 𝑀)
10		df-epi 17747	. . . . . 6 ⊢ Epi = (𝑐 ∈ Cat ↦ tpos (Mono‘(oppCat‘𝑐)))
11	7	fvexi 6915	. . . . . . 7 ⊢ 𝑀 ∈ V
12	11	tposex 8275	. . . . . 6 ⊢ tpos 𝑀 ∈ V
13	9, 10, 12	fvmpt 7009	. . . . 5 ⊢ (𝐶 ∈ Cat → (Epi‘𝐶) = tpos 𝑀)
14	2, 13	syl 17	. . . 4 ⊢ (𝜑 → (Epi‘𝐶) = tpos 𝑀)
15	1, 14	eqtrid 2778	. . 3 ⊢ (𝜑 → 𝐸 = tpos 𝑀)
16	15	oveqd 7441	. 2 ⊢ (𝜑 → (𝑌𝐸𝑋) = (𝑌tpos 𝑀𝑋))
17		ovtpos 8256	. 2 ⊢ (𝑌tpos 𝑀𝑋) = (𝑋𝑀𝑌)
18	16, 17	eqtr2di 2783	1 ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑌𝐸𝑋))

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  ‘cfv 6554  (class class class)co 7424  tpos ctpos 8240  Catccat 17677  oppCatcoppc 17724  Monocmon 17744  Epicepi 17745

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-fv 6562  df-ov 7427  df-tpos 8241  df-epi 17747

This theorem is referenced by:  oppcepi  17755  isepi  17756  epii  17759  sectepi  17800  episect  17801  fthepi  17950