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Theorem oppcepi 17690

Description: An epimorphism in the opposite category is a monomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)

**Hypotheses**
Ref	Expression
oppcmon.o	⊢ 𝑂 = (oppCat‘𝐶)
oppcmon.c	⊢ (𝜑 → 𝐶 ∈ Cat)
oppcepi.e	⊢ 𝐸 = (Epi‘𝑂)
oppcepi.m	⊢ 𝑀 = (Mono‘𝐶)

**Assertion**
Ref	Expression
oppcepi	⊢ (𝜑 → (𝑋𝐸𝑌) = (𝑌𝑀𝑋))

**Proof of Theorem oppcepi**
Step	Hyp	Ref	Expression
1		oppcepi.m	. . . 4 ⊢ 𝑀 = (Mono‘𝐶)
2		oppcmon.o	. . . . . . 7 ⊢ 𝑂 = (oppCat‘𝐶)
3	2	2oppchomf 17674	. . . . . 6 ⊢ (Hom_f ‘𝐶) = (Hom_f ‘(oppCat‘𝑂))
4	3	a1i 11	. . . . 5 ⊢ (𝜑 → (Hom_f ‘𝐶) = (Hom_f ‘(oppCat‘𝑂)))
5	2	2oppccomf 17675	. . . . . 6 ⊢ (comp_f‘𝐶) = (comp_f‘(oppCat‘𝑂))
6	5	a1i 11	. . . . 5 ⊢ (𝜑 → (comp_f‘𝐶) = (comp_f‘(oppCat‘𝑂)))
7		oppcmon.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
8	2	oppccat 17672	. . . . . . 7 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
9	7, 8	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑂 ∈ Cat)
10		eqid 2732	. . . . . . 7 ⊢ (oppCat‘𝑂) = (oppCat‘𝑂)
11	10	oppccat 17672	. . . . . 6 ⊢ (𝑂 ∈ Cat → (oppCat‘𝑂) ∈ Cat)
12	9, 11	syl 17	. . . . 5 ⊢ (𝜑 → (oppCat‘𝑂) ∈ Cat)
13	4, 6, 7, 12	monpropd 17688	. . . 4 ⊢ (𝜑 → (Mono‘𝐶) = (Mono‘(oppCat‘𝑂)))
14	1, 13	eqtrid 2784	. . 3 ⊢ (𝜑 → 𝑀 = (Mono‘(oppCat‘𝑂)))
15	14	oveqd 7428	. 2 ⊢ (𝜑 → (𝑌𝑀𝑋) = (𝑌(Mono‘(oppCat‘𝑂))𝑋))
16		eqid 2732	. . 3 ⊢ (Mono‘(oppCat‘𝑂)) = (Mono‘(oppCat‘𝑂))
17		oppcepi.e	. . 3 ⊢ 𝐸 = (Epi‘𝑂)
18	10, 9, 16, 17	oppcmon 17689	. 2 ⊢ (𝜑 → (𝑌(Mono‘(oppCat‘𝑂))𝑋) = (𝑋𝐸𝑌))
19	15, 18	eqtr2d 2773	1 ⊢ (𝜑 → (𝑋𝐸𝑌) = (𝑌𝑀𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ‘cfv 6543 (class class class)co 7411 Catccat 17612 Hom_f chomf 17614 comp_fccomf 17615 oppCatcoppc 17659 Monocmon 17679 Epicepi 17680

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-tpos 8213 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-hom 17225 df-cco 17226 df-cat 17616 df-cid 17617 df-homf 17618 df-comf 17619 df-oppc 17660 df-mon 17681 df-epi 17682

This theorem is referenced by: (None)

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