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 6863	. . . . . . . . . 10 ⊢ (𝑐 = 𝐶 → (oppCat‘𝑐) = (oppCat‘𝐶))
4		oppcmon.o	. . . . . . . . . 10 ⊢ 𝑂 = (oppCat‘𝐶)
5	3, 4	eqtr4di 2814	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (oppCat‘𝑐) = 𝑂)
6	5	fveq2d 6867	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Mono‘(oppCat‘𝑐)) = (Mono‘𝑂))
7		oppcmon.m	. . . . . . . 8 ⊢ 𝑀 = (Mono‘𝑂)
8	6, 7	eqtr4di 2814	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Mono‘(oppCat‘𝑐)) = 𝑀)
9	8	tposeqd 8204	. . . . . 6 ⊢ (𝑐 = 𝐶 → tpos (Mono‘(oppCat‘𝑐)) = tpos 𝑀)
10		df-epi 17747	. . . . . 6 ⊢ Epi = (𝑐 ∈ Cat ↦ tpos (Mono‘(oppCat‘𝑐)))
11	7	fvexi 6877	. . . . . . 7 ⊢ 𝑀 ∈ V
12	11	tposex 8235	. . . . . 6 ⊢ tpos 𝑀 ∈ V
13	9, 10, 12	fvmpt 6971	. . . . 5 ⊢ (𝐶 ∈ Cat → (Epi‘𝐶) = tpos 𝑀)
14	2, 13	syl 17	. . . 4 ⊢ (𝜑 → (Epi‘𝐶) = tpos 𝑀)
15	1, 14	eqtrid 2808	. . 3 ⊢ (𝜑 → 𝐸 = tpos 𝑀)
16	15	oveqd 7409	. 2 ⊢ (𝜑 → (𝑌𝐸𝑋) = (𝑌tpos 𝑀𝑋))
17		ovtpos 8216	. 2 ⊢ (𝑌tpos 𝑀𝑋) = (𝑋𝑀𝑌)
18	16, 17	eqtr2di 2813	1 ⊢ (𝜑 → (𝑋𝑀𝑌) = (𝑌𝐸𝑋))

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  ‘cfv 6517  (class class class)co 7392  tpos ctpos 8200  Catccat 17679  oppCatcoppc 17726  Monocmon 17744  Epicepi 17745

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-fv 6525  df-ov 7395  df-tpos 8201  df-epi 17747

This theorem is referenced by:  oppcepi  17755  isepi  17756  epii  17759  sectepi  17800  episect  17801  fthepi  17946